07.06.26 · representation-theory / lie-algebraic

$G_{2}$ via the octonions

shipped3 tiersLean: none

Anchor (Master): Fulton-Harris *Representation Theory* §22 in full; Adams 1996 *Lectures on Exceptional Lie Groups* Ch. 5; Jacobson 1958 *Composition algebras and their automorphisms* (Rend. Circ. Mat. Palermo 7); Cartan 1914 *Les groupes réels simples finis et continus* (Ann. Sci. ENS 31) — the real forms of $\mathfrak{g}_2$

Intuition Beginner

There are exactly four number systems in which you can add, subtract, multiply, divide, and measure length so that the length of a product is the product of the lengths. They are the real numbers, the complex numbers, the quaternions, and the octonions. Each is built from the one before by doubling its dimension: 1, then 2, then 4, then 8. The octonions are the last and largest; there is no sixteen-dimensional sequel.

The octonions are written $O$ . An octonion is a list of eight real numbers, just as a complex number is a list of two. They multiply by a fixed table of rules. The multiplication is strange: order matters ( $ab$ need not equal $ba$ ), and even grouping matters a little ( $(ab) c$ need not equal $a (b c)$ ). Yet every nonzero octonion has a multiplicative inverse, so you can always divide.

A natural question: what are the symmetries of this multiplication table? A symmetry is a way of relabelling the eight coordinates that leaves every product unchanged — if $ab = c$ before relabelling, then the relabelled $a$ times the relabelled $b$ equals the relabelled $c$ . Such symmetries form a group.

That group is called $G_{2}$ . It is a compact Lie group of dimension fourteen, and it is the smallest of the five exceptional Lie groups — the five symmetry groups that do not fit into any of the familiar infinite families of rotations, unitary maps, or symplectic maps. This unit builds $G_{2}$ from the octonions and reads its hidden root-system skeleton straight off the multiplication table.

Visual Beginner

Picture the seven imaginary octonion directions $e_{1}, \dots, e_{7}$ as the seven points of a tiny diagram called the Fano plane: a triangle with its three midpoints, its inscribed circle, and the three medians drawn in. Seven points, seven lines, with exactly three points on every line and three lines through every point. Each line, taken with an arrow, records one multiplication rule: if a line runs $e_{i} \to e_{j} \to e_{k}$ , then $e_{i} e_{j} = e_{k}$ .

Reading the picture: pick any oriented line, read its three points in cyclic order, and you get a product rule together with its cyclic shifts. Reversing the arrow flips the sign. Every product of two distinct imaginary units is read off this way, and the seven squares are all $- 1$ . The continuous symmetries of this whole oriented structure assemble into $G_{2}$ .

Worked example Beginner

Take two specific imaginary octonions and multiply them, then check that the length rule holds.

Use the standard labelling in which the seven oriented Fano lines include $e_{1} e_{2} = e_{3}$ , $e_{1} e_{4} = e_{5}$ , and $e_{2} e_{4} = e_{6}$ . Suppose we want $e_{3} e_{5}$ . The Fano line through $e_{3}$ and $e_{5}$ also contains $e_{6}$ , oriented so that $e_{3} e_{5} = e_{6}$ . So the product of two unit-length imaginary octonions is again a unit-length imaginary octonion. The length of the product is $1$ , and the product of the two lengths is $1 \cdot 1 = 1$ . The length rule holds.

Now multiply in the other order: $e_{5} e_{3}$ . Reversing the order of two distinct imaginary units flips the sign, so $e_{5} e_{3} = - e_{6}$ . The two answers differ by a sign — concrete evidence that octonion multiplication does not commute.

A symmetry of the octonions must respect all of this at once. If a relabelling sends $e_{3} \mapsto e_{3}^{'}$ , $e_{5} \mapsto e_{5}^{'}$ , and $e_{6} \mapsto e_{6}^{'}$ , then it is only allowed if $e_{3}^{'} e_{5}^{'} = e_{6}^{'}$ in the new labels too. The fourteen independent "infinitesimal relabellings" that respect every such rule are what we will count.

Check your understanding Beginner

Formal definition Intermediate+

Let $H$ denote the quaternions. The octonions $O$ are obtained from $H$ by the Cayley-Dickson construction: as a real vector space $O = H \oplus H$ , with conjugation, multiplication, and norm defined by

\overline{(a, b)} = (\overset{a}{ˉ}, - b), (a, b) (c, d) = (a c - \overset{ˉ}{d} b, d a + b \overset{c}{ˉ}), N (a, b) = N (a) + N (b),

where on the right $N (a) = a \overset{a}{ˉ}$ is the quaternion norm. This makes $O$ an $8$ -dimensional real algebra with unit $1 = (1, 0)$ . The construction is one stage of the chain $R \to C \to H \to O$ , each doubling dimension and shedding one structural property: $C$ loses self-conjugacy, $H$ loses commutativity, and $O$ loses associativity.

Composition algebra. The quadratic form $N (x) = x \overset{x}{ˉ}$ is multiplicative: $N (x y) = N (x) N (y)$ for all $x, y \in O$ , equivalently $∣ x y ∣ = ∣ x ∣∣ y ∣$ for the associated norm $∣ x ∣ = N (x)$ . An algebra with identity carrying a non-degenerate multiplicative quadratic form is a composition algebra. Hurwitz's theorem states that the only real composition algebras with positive-definite norm are $R, C, H, O$ . Because $N$ is positive-definite and multiplicative, every nonzero $x$ has inverse $x^{- 1} = \overset{x}{ˉ} / N (x)$ , so $O$ is a division algebra.

Alternativity and the Moufang identities. Octonion multiplication is not associative, but it is alternative: the associator $[x, y, z] := (x y) z - x (y z)$ is an alternating function of its three arguments. Equivalently, any two elements generate an associative subalgebra. Alternativity is sharpened by the three Moufang identities

(x y) (z x) = x ((y z) x), (x (y x)) z = x (y (x z)), ((x y) x) z = x (y (x z)),

which control all the ways products of three or four octonions can be reassociated.

Imaginary octonions. Write $x = Re (x) + Im (x)$ with $Re (x) = \frac{1}{2} (x + \overset{x}{ˉ}) \cdot 1$ . The imaginary octonions $Im O = {x : \overset{x}{ˉ} = - x}$ form a $7$ -dimensional real inner-product space, spanned by $e_{1}, \dots, e_{7}$ with $e_{i} e_{j} = - δ_{ij} + \sum_{k} c_{ij k} e_{k}$ , where the totally antisymmetric structure constants $c_{ij k}$ are the seven oriented Fano lines.

Definition ( $G_{2}$ and $g_{2}$ ). The compact group $G_{2}$ is the automorphism group of the octonion algebra, $$ G_2 := \mathrm{Aut}(\mathbb{O}) = { g \in \mathrm{GL}(\mathbb{O}) : g(xy) = g(x), g(y)\ \text{for all } x, y}. $$ Its Lie algebra is the derivation algebra $$ \mathfrak{g}_2 := \mathrm{Der}(\mathbb{O}) = { D \in \mathrm{End}(\mathbb{O}) : D(xy) = D(x), y + x, D(y)}. $$ Every automorphism fixes the unit $1$ and preserves the norm, so $G_{2}$ fixes $R \cdot 1$ and acts on the orthogonal complement $Im O ≅ R^{7}$ ; this is the $7$ -dimensional fundamental representation.

Key theorem with proof Intermediate+

Theorem ( $g_{2} = Der (O)$ is $14$ -dimensional, simple, of type $G_{2}$ ). The derivation algebra $g_{2} = Der (O)$ is a compact real simple Lie algebra of dimension $14$ and rank $2$ , with complexification $g_{2} (C)$ the exceptional simple Lie algebra of type $G_{2}$ . It embeds into $so_{7}$ as the stabiliser of a generic alternating $3$ -form on $R^{7} = Im O$ .

Proof. A derivation $D$ satisfies $D (1) = D (1 \cdot 1) = 2 D (1)$ , so $D (1) = 0$ and $D$ preserves $Im O$ . Differentiating $N (x) = x \overset{x}{ˉ}$ along the one-parameter group $exp (t D)$ shows $⟨ D x, y ⟩ + ⟨ x, D y ⟩ = 0$ , so $Der (O)$ embeds into $so (Im O) = so_{7}$ , which has dimension $21$ .

Now count constraints. On $Im O$ the product splits as $x y = - ⟨ x, y ⟩ + x \times y$ , where the cross product $x \times y = Im (x y)$ is the imaginary part. A skew map $D \in so_{7}$ is a derivation iff it respects this cross product: $D (x \times y) = D x \times y + x \times D y$ . The cross product is encoded by the alternating $3$ -form $ϕ (x, y, z) := ⟨ x \times y, z ⟩ = ⟨ x y, z ⟩$ , and $D$ is a derivation iff $D$ annihilates $ϕ$ , i.e. $ϕ (D x, y, z) + ϕ (x, D y, z) + ϕ (x, y, D z) = 0$ . So $Der (O) = so_{7} \cap Stab (ϕ)$ .

The space of alternating $3$ -forms on $R^{7}$ has dimension $(3 7) = 35$ . The group $GL_{7} (R)$ , of dimension $49$ , acts on this $35$ -dimensional space, and $ϕ$ lies in a generic open orbit; its $GL_{7}$ -stabiliser therefore has dimension $49 - 35 = 14$ . Intersecting with the orthogonal condition is automatic because a $3$ -form stabiliser of this type already preserves the metric it determines. Hence $dim Der (O) = 14$ . Rank $2$ and simplicity follow because the only compact simple Lie algebra of dimension $14$ in the Cartan-Weyl list (07.04.01) is $g_{2}$ , whose complexification is the exceptional type- $G_{2}$ algebra; the alternative is $so_{7}$ of dimension $21$ or products of smaller algebras, none of dimension $14$ . $□$

Bridge. This construction builds toward the rest of exceptional Lie theory and appears again in the Freudenthal magic square, where $g_{2} = Der (O)$ is the corner from which $f_{4}, e_{6}, e_{7}, e_{8}$ are generated by feeding the octonions into Jordan algebras. The foundational reason $G_{2}$ has dimension $14$ is exactly the orbit count $49 - 35 = 14$ for the generic $3$ -form, and this is dual to the spinor-stabiliser count in 03.09.13, where the same number arises as $dim Spin (7) - 7 = 21 - 7$ ; the two derivations of $14$ are the same group seen through two windows. The central insight is that an exceptional symmetry is the symmetry of a non-associative algebra, and this generalises: every exceptional Lie group is the symmetry group of an octonionic structure. Putting these together, the bridge is the identification $g_{2} (C) \subset so_{7} (C)$ , which lets the full machinery of the classical root-system theory be turned on the smallest exceptional algebra.

Exercises Intermediate+

Exercise 3 (medium, short answer).

The $7$ -dimensional representation of $G_{2}$ has weights that are the six short roots of $G_{2}$ together with the zero weight. List how many nonzero weights there are and explain why the zero weight occurs once.

Hint

$dim = 7$ . The short roots come in $\pm$ pairs; the remaining weight is forced by traceless skew action.

Answer

There are $6$ nonzero weights — the $6$ short roots of $G_{2}$ , in three $\pm$ pairs — plus the zero weight with multiplicity one, giving $6 + 1 = 7$ . The zero weight occurs because the Cartan subalgebra (a maximal torus of $G_{2}$ ) acts on $Im O$ by a traceless skew transformation, and a $7$ -dimensional real representation of a rank- $2$ torus with weights in $\pm$ pairs must carry one fixed line. Concretely, that fixed line is spanned by the imaginary unit left invariant by the chosen maximal torus.

Exercise 5 (hard, proof).

Prove that $Der (O)$ acts irreducibly on $Im O$ .

Hint

An invariant subspace closed under the cross product and orthogonal complement would force a composition subalgebra of forbidden dimension.

Answer

Suppose $W \subset Im O$ is a nonzero $Der (O)$ -invariant subspace. The transitivity of $G_{2} = Aut (O)$ on the unit sphere of $Im O$ (which follows because any two imaginary units of norm $1$ generate isomorphic copies of $C \subset O$ and an automorphism carries one to the other) shows the $G_{2}$ -orbit of any unit vector is all of $S^{6}$ . An invariant subspace containing one unit vector therefore contains the whole sphere, hence all of $Im O$ . So $W = Im O$ and the representation is irreducible. (Equivalently: the $7$ is a fundamental irreducible with highest weight the dominant short root, so it admits no proper invariant subspace.) $□$

Exercise 6 (hard, conceptual).

The two real forms of $g_{2} (C)$ are the compact form $Der (O)$ and the split form. Identify the algebra whose derivations give the split real form.

Hint

Replace the positive-definite norm by an indefinite one — the split octonions.

Answer

The split real form $g_{2}^{(2)}$ (signature $- 14 + rank$ , i.e. the non-compact split form) is the derivation algebra $Der (O_{s})$ of the split octonions $O_{s}$ — the $8$ -dimensional composition algebra whose norm form has signature $(4, 4)$ instead of $(8, 0)$ . The split octonions contain zero divisors (the norm is isotropic), so they are not a division algebra, but their automorphism group is the split real form of $G_{2}$ . Jacobson 1958 ^{[Jacobson 1958]} classified composition algebras and matched their automorphism groups to the two real forms of $G_{2}$ ; Cartan 1914 ^{[Cartan 1914]} had already found both real forms by Lie-theoretic means.

Advanced results Master

The $7$ -dimensional representation and the invariant $3$ -form. The fundamental $7$ -dimensional representation $V_{7} = Im O$ carries a $G_{2}$ -invariant alternating $3$ -form $ϕ (x, y, z) = ⟨ x y, z ⟩$ and the induced metric; conversely, $G_{2} = Stab_{GL_{7}} (ϕ)$ . In the standard coordinates dual to the Fano basis, $$ \phi = e^{123} + e^{145} + e^{167} + e^{246} - e^{257} - e^{347} - e^{356}, $$ where $e^{ij k} = e^{i} \land e^{j} \land e^{k}$ and the seven terms are the seven oriented Fano lines. The Hodge dual $* ϕ$ is the $G_{2}$ -invariant $4$ -form governing coassociative geometry; both forms are parallel on a $G_{2}$ -holonomy $7$ -manifold.

Reading the root system off the multiplication table. The maximal torus $T \subset G_{2}$ has rank $2$ ; choose it to act on $Im O$ fixing $e_{7}$ and rotating the two-planes $⟨ e_{1}, e_{2} ⟩$ , $⟨ e_{3}, e_{4} ⟩$ , $⟨ e_{5}, e_{6} ⟩$ by angles $θ_{1}, θ_{2}, θ_{1} + θ_{2}$ respectively (the angles are constrained because the products $e_{1} e_{3}, e_{1} e_{5}, e_{3} e_{5}$ must be preserved). The three planes carry weights $\pm α, \pm β, \pm (α + β)$ : these are the six short roots of $G_{2}$ , and they are exactly the weights of the $7$ (with $e_{7}$ giving the zero weight). The adjoint representation $g_{2}$ then has weights the $12$ roots, $6$ short plus $6$ long, with the long roots $\pm (2 α + β), \pm (α + 2 β), \pm (α - β)$ arising from the bracket. The two root lengths, in ratio $3 : 1$ , are the signature of the triple bond in the $G_{2}$ Dynkin diagram (07.04.01).

The complex form $g_{2} (C) \subset so_{7} (C)$ . Complexifying, $g_{2} (C)$ is the $14$ -dimensional subalgebra of $so_{7} (C)$ preserving both the complexified inner product and the complexified $3$ -form. The branching of the adjoint $so_{7} (C)$ -module ( $dim 21$ ) under $g_{2} (C)$ is $21 = 14 \oplus 7$ : the derivation algebra plus a complementary copy of the $7$ , the latter being the infinitesimal motions of $SO_{7}$ that change the $3$ -form. This $21 = 14 \oplus 7$ split is the algebraic shadow of $dim G_{2} = dim SO_{7} - 7$ .

Synthesis. The octonion construction is the foundational reason $G_{2}$ exists at all: it is exactly the automorphism group of the unique $8$ -dimensional real division algebra, and its $14$ dimensions are the codimension of the generic $3$ -form orbit in $Λ^{3} (R^{7})^{*}$ . The central insight is that the entire root-system data of the smallest exceptional algebra — two root lengths, six short roots, six long roots — is encoded combinatorially in the octonion multiplication table, with the short roots reading off as the torus weights of the imaginary octonions and the long roots as their brackets. This is exactly the same group that 03.09.13 constructs by stabilising a spinor in $Spin (7)$ , and the bridge between the two constructions is the identification of the invariant $3$ -form with a spinor bilinear, so the derivation-algebra picture is dual to the spinor-stabiliser picture. Putting these together, the inclusion $g_{2} (C) \subset so_{7} (C)$ with branching $21 = 14 \oplus 7$ generalises into the Freudenthal magic square, where octonionic Jordan algebras manufacture $f_{4}, e_{6}, e_{7}, e_{8}$ by the same recipe of feeding $O$ into a derivation count; $G_{2}$ is the first and smallest instance of the principle that exceptional symmetry is octonionic symmetry.

Full proof set Master

Proposition (transitivity of $G_{2}$ on the unit sphere $S^{6} \subset Im O$ , and the resulting homogeneous space). $G_{2}$ acts transitively on the unit sphere of $Im O$ with stabiliser $SU (3)$ , so $S^{6} = G_{2} / SU (3)$ .

Proof. Let $u, u^{'} \in Im O$ be unit imaginary octonions. Each generates a copy of the complex numbers: $R [u] ≅ C$ with $u^{2} = - 1$ , and likewise $R [u^{'}] ≅ C$ . Choosing a unit imaginary octonion $v$ orthogonal to $u$ and to $1$ , the subalgebra generated by $u$ and $v$ is a copy of the quaternions $H$ (alternativity guarantees that two elements generate an associative subalgebra, and the norm pins down the relations). Hence any unit imaginary octonion sits inside a quaternion subalgebra, and any two such are conjugate under an automorphism that extends the obvious isomorphism of the generated subalgebras. This gives transitivity on $S^{6}$ .

For the stabiliser, fix $u = e_{7}$ . An automorphism fixing $e_{7}$ preserves its centraliser-type decomposition $Im O = R e_{7} \oplus W$ , where $W = e_{7}^{⊥} \cap Im O$ is $6$ -dimensional. Left multiplication by $e_{7}$ on $W$ satisfies $(L_{e_{7}})^{2} = - 1$ , so it is a complex structure $J$ on $W$ , making $W ≅ C^{3}$ . An automorphism fixing $e_{7}$ commutes with $J$ and preserves the Hermitian form; one checks it also preserves the induced complex volume form, so it lies in $SU (3)$ , and conversely every element of $SU (3)$ arises. Therefore $Stab_{G_{2}} (e_{7}) = SU (3)$ .

Dimension check: $dim G_{2} - dim SU (3) = 14 - 8 = 6 = dim S^{6}$ , consistent with transitivity. $□$

Corollary (the $7$ restricts correctly). Under $SU (3) \subset G_{2}$ , the $7$ -dimensional representation branches as $7 = 1 \oplus 3 \oplus \overset{ˉ}{3}$ : the fixed line $R e_{7}$ complexifies to the one-dimensional $1$ , and $W ≅ C^{3}$ carries the standard $3$ and its conjugate $\overset{ˉ}{3}$ . This branching is the representation-theoretic fingerprint of the homogeneous space $S^{6} = G_{2} / SU (3)$ and of the almost-complex structure $S^{6}$ inherits from the octonions.

Connections Master

The Cartan-Weyl classification 07.04.01 supplies the verdict that fixes $Der (O)$ as type $G_{2}$ : it is the unique compact simple Lie algebra of dimension $14$ and rank $2$ . Without that classification the dimension count $14$ would be a number; with it, the octonion construction lands exactly on the first exceptional algebra, and the two root lengths become the triple bond of the $G_{2}$ Dynkin diagram catalogued there.
The root system 07.06.03 of type $G_{2}$ — six long and six short roots, length ratio $3$ , Weyl group the dihedral group of order $12$ — is read directly off the octonion multiplication table here: the short roots are the maximal-torus weights on $Im O$ , and the long roots are their brackets. This unit gives the most concrete realisation of an abstract $G_{2}$ root diagram, turning the combinatorics of 07.06.03 into the geometry of a division algebra.
The triality unit 03.09.13 constructs the same group $G_{2} = Aut (O)$ by the complementary spinor route: it builds the octonions from $Spin (8)$ half-spinor squaring and obtains $G_{2}$ as the stabiliser of a unit spinor inside $Spin (7)$ . This unit instead builds $G_{2}$ as a derivation algebra and reads its root system off the multiplication table. The bridge between the two is the identification of the $G_{2}$ -invariant $3$ -form with a spinor bilinear, recovered in Exercise 4; together the two units present one group through two genuinely different windows.
The representations of $sl_{3} C$ 07.06.12 supply the ambient $SU (3) \subset G_{2}$ whose standard $3$ and conjugate $\overset{ˉ}{3}$ appear in the branching $7 = 1 \oplus 3 \oplus \overset{ˉ}{3}$ of the Full proof set, fixing the homogeneous-space structure $S^{6} = G_{2} / SU (3)$ .

Historical & philosophical context Master

The exceptional algebra of type $G_{2}$ was the first piece of the exceptional Lie family to appear, and it appeared twice in 1893. Wilhelm Killing's classification programme had predicted a $14$ -dimensional simple algebra outside the four classical families; Friedrich Engel and Élie Cartan, working independently, gave the first concrete realisations that year. Cartan's $1894$ thesis placed $g_{2} (C)$ firmly in the classification, and his $1914$ paper Les groupes réels simples finis et continus ^{[Cartan 1914]} determined its two real forms — the compact one and the split one — by purely Lie-theoretic means, before any octonionic interpretation was attached to them.

The octonions themselves were older, discovered by Arthur Cayley in $1845$ (and independently by John Graves slightly earlier), as the natural eight-dimensional successor to Hamilton's quaternions. For decades the $14$ -dimensional algebra and the octonions led separate lives. The unification is due to the recognition — crystallised by Cartan and made structural by Nathan Jacobson's $1958$ study of composition algebras ^{[Jacobson 1958]} — that $g_{2} = Der (O)$ and $G_{2} = Aut (O)$ : the abstractly-classified exceptional algebra is nothing other than the infinitesimal symmetry of the largest division algebra. Hurwitz's $1898$ theorem, pinning the composition algebras to exactly $R, C, H, O$ , explains why there is one and only one such smallest exceptional group.

Philosophically, $G_{2}$ is the cleanest answer to a recurring question in mathematics: are the exceptional structures accidents, or are they forced? The octonion construction argues for forced. The chain of normed division algebras terminates at dimension $8$ for a structural reason (multiplicativity of the norm fails past $O$ ), and the symmetries of that terminal algebra are necessarily exceptional because no classical family has the right dimension. Adams's posthumous Lectures on Exceptional Lie Groups makes this the organising thesis of all five exceptional groups: each is the symmetry group of an octonionic object. $G_{2}$ , the symmetry of $O$ itself, is where that thesis begins.

Bibliography Master

Fulton, W. & Harris, J., Representation Theory: A First Course, Graduate Texts in Mathematics 129, Springer, 1991. §22 — the octonions, $g_{2} = Der (O)$ , the $7$ -dimensional representation, the type- $G_{2}$ root system.
Baez, J. C., "The octonions", Bull. AMS 39 (2002), 145–205. §§1–4 — Cayley-Dickson construction, Hurwitz's theorem, $G_{2} = Aut (O)$ .
Jacobson, N., "Composition algebras and their automorphisms", Rend. Circ. Mat. Palermo 7 (1958), 55–80. The two real forms of $G_{2}$ as automorphism groups of the compact and split octonions.
Cartan, É., "Les groupes réels simples finis et continus", Ann. Sci. École Norm. Sup. 31 (1914), 263–355. The real forms of the exceptional simple Lie algebras, including $g_{2}$ .
Cartan, É., Sur la structure des groupes de transformations finis et continus (Thèse), Paris, 1894. Places $g_{2} (C)$ in the classification of complex simple Lie algebras.
Adams, J. F., Lectures on Exceptional Lie Groups, Chicago Lectures in Mathematics, University of Chicago Press, 1996. Ch. 5 — $G_{2}$ via the octonions.
Hurwitz, A., "Über die Composition der quadratischen Formen von beliebig vielen Variablen", Nachr. Ges. Wiss. Göttingen (1898), 309–316. The four real composition algebras.
Harvey, F. R., Spinors and Calibrations, Academic Press, 1990. The $G_{2}$ -invariant $3$ -form and its spinorial origin.

Prerequisites

07.04.01
07.06.03
03.09.13

Tier anchors

beginner: Baez 2002 *The octonions* (Bull. AMS 39) §1-2 (the four normed division algebras, Cayley-Dickson doubling, the Fano-plane multiplication table); Stillwell *Naive Lie Theory* Ch. 8 informal account of symmetry groups of algebras
intermediate: Fulton-Harris *Representation Theory* §22.1-22.3 (the octonion algebra, $\mathfrak{g}_2 = \mathrm{Der}(\mathbb{O})$, the 7-dimensional representation on $\mathrm{Im}\,\mathbb{O}$, the type-$G_2$ root system); Baez 2002 §4 ($G_2 = \mathrm{Aut}(\mathbb{O})$)
master: Fulton-Harris *Representation Theory* §22 in full; Adams 1996 *Lectures on Exceptional Lie Groups* Ch. 5; Jacobson 1958 *Composition algebras and their automorphisms* (Rend. Circ. Mat. Palermo 7); Cartan 1914 *Les groupes réels simples finis et continus* (Ann. Sci. ENS 31) — the real forms of $\mathfrak{g}_2$

References

Fulton-Harris — Representation Theory · §22 The exceptional Lie algebras; §22.1 the octonions and $\mathfrak{g}_2 = \mathrm{Der}(\mathbb{O})$; §22.2 the 7-dimensional representation and the root system; §22.3 $\mathfrak{g}_2 \subset \mathfrak{so}_7$
Baez — The octonions (Bull. AMS 39, 2002) · §1-2 (Cayley-Dickson construction, normed division algebras, Hurwitz); §4 ($G_2$ as the automorphism group of $\mathbb{O}$, dimension count $14$)
Jacobson — Composition algebras and their automorphisms (Rend. Circ. Mat. Palermo 7, 1958) · the structure theory of composition algebras and the identification of $\mathrm{Aut}$ of the split and compact octonions with the two real forms of $G_2$
Cartan — Les groupes réels simples finis et continus (Ann. Sci. ENS 31, 1914) · the classification of the real forms of the exceptional complex simple Lie algebra $\mathfrak{g}_2(\mathbb{C})$ — compact and split
Adams — Lectures on Exceptional Lie Groups · Ch. 5 ($G_2$ as $\mathrm{Aut}(\mathbb{O})$, the 7-dimensional representation, the inclusion into $\mathrm{Spin}(7)$)

Estimated time

beginner: 20m
intermediate: 50m
master: 85m