The Campbell–Baker–Hausdorff formula

Intuition Beginner

When two numbers are added, the order does not matter, and the exponential of a sum splits cleanly: $e^{a+b}=e^a{e}^b$. For matrices, or for the flows that a Lie group runs, this splitting fails. Multiplying $e^X$ by $e^Y$ does not give $e^{X+Y}$ unless $X$ and $Y$ commute. The product is still the exponential of something, written $e^Z$. The Campbell–Baker–Hausdorff formula tells us exactly what that $Z$ is.

The first surprise is that $Z$ is not just $X+Y$. It starts there, but it picks up a correction $\frac{1}{2}[X,Y]$ that measures how badly $X$ and $Y$ fail to commute, and then smaller corrections built from nested brackets.

The deeper surprise is the kind of object $Z$ turns out to be. Every term is a bracket of $X$ and $Y$ — never a plain product like $XY$. The whole answer lives inside the bracket structure, which is the Lie algebra. That single fact is why a Lie group is encoded in its Lie algebra.

Visual Beginner

Picture two short flows on a curved surface. One flow pushes you in the $X$ direction for one second; the other pushes you in the $Y$ direction for one second. Running the first and then the second lands you at a point. Running them as a single combined flow in direction $Z$ must land you at the very same point.

The combined direction $Z$ is close to $X+Y$, but tilted. The tilt is the bracket $[X,Y]$, the same object that measures the small gap between doing $X$-then-$Y$ and doing $Y$-then-$X$. When the two flows commute, there is no gap, no tilt, and $Z$ is just $X+Y$.

Worked example Beginner

Take the two matrices $$ X=\begin{pmatrix}0&1\0&0\end{pmatrix},\qquad Y=\begin{pmatrix}0&0\1&0\end{pmatrix}. $$ Their bracket is $[X,Y]=XY-YX=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$, which is not zero, so they do not commute.

Step 1. Read off the first two terms. The formula gives $Z=X+Y+\frac{1}{2}[X,Y]+\cdots$. The leading piece $X+Y=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$.

Step 2. Add the bracket correction. Half of $[X,Y]$ is $\left(\begin{array}{cc}\frac{1}{2}& 0\\ 0& -\frac{1}{2}\end{array}\right)$, so through second order $Z\approx \left(\begin{array}{cc}\frac{1}{2}& 1\\ 1& -\frac{1}{2}\end{array}\right)$.

Step 3. Sanity check on a commuting pair. If instead $X$ and $Y$ were both diagonal, every bracket would be zero, the correction would vanish, and $Z$ would equal $X+Y$ exactly — matching the ordinary rule $e^a{e}^b=e^{a+b}$.

What this shows: the bracket correction is the entire content of the formula. When brackets vanish, the formula collapses to plain addition.

Check your understanding Beginner

Formal definition Intermediate+

Let $\mathfrak{g}$ be a Lie algebra over a field of characteristic zero, and work inside the completed free associative algebra on two generators — equivalently, treat $X,Y$ as formal noncommuting variables and pass to the completion $\hat{A}=\mathbb{Q}⟨⟨X,Y⟩⟩$ in which power series converge by total degree. The exponential and logarithm are defined formally, $$ e^{W}=\sum_{k\ge 0}\frac{W^{k}}{k!},\qquad \log(1+U)=\sum_{k\ge 1}\frac{(-1)^{k-1}}{k}U^{k}, $$ for $W$ and $U$ of positive degree.

The Campbell–Baker–Hausdorff series is $$ Z(X,Y)=\log\bigl(e^{X}e^{Y}\bigr)\in\widehat{A}, $$ the unique element with $e^X{e}^Y=e^{Z(X,Y)}$. Its low-degree terms are $$ Z=X+Y+\tfrac12[X,Y]+\tfrac1{12}\bigl([X,[X,Y]]+[Y,[Y,X]]\bigr)-\tfrac1{24}[Y,[X,[X,Y]]]+\cdots, $$ where $[A,B]=AB-BA$.

Definition (Lie element). An element of $\hat{A}$ is a Lie element if it lies in the closed Lie subalgebra generated by $X$ and $Y$ under the commutator bracket — that is, if it is an (infinite) sum of iterated brackets. The free Lie algebra $\hat{L}(X,Y)$ 07.06.13 is exactly this subalgebra.

The central structural claim, proved below, is that $Z(X,Y)$ is a Lie element: although $\log (e^X{e}^Y)$ is defined through associative products, every coefficient reorganises into brackets.

Counterexamples to common slips

• $Z$ is not $X+Y+\frac{1}{2}[X,Y]$ exactly. That is only the degree-$\le 2$ truncation. The degree-$3$ terms $\frac{1}{12}([X,[X,Y]]+[Y,[Y,X]])$ are genuinely present; dropping them is the source of second-order splitting errors in numerical integrators.
• The series is not symmetric in $X,Y$. Swapping $X\leftrightarrow Y$ sends $Z(X,Y)$ to $Z(Y,X)=\log (e^Y{e}^X)$, and $[X,Y]=-[Y,X]$ flips the sign of the degree-$2$ term. The symmetric variant $\log (e^{X/2}{e}^Y{e}^{X/2})$ has no even-degree-in-$Y$ bracket of first order, which is why Strang splitting is second-order accurate.
• Convergence is not automatic. Over $\mathbb{R}$ or $\mathbb{C}$ the series converges only for $\Vert X\Vert ,\Vert Y\Vert$ small; the bound $\Vert X\Vert +\Vert Y\Vert <\log 2$ guarantees it. As a formal series in $\hat{A}$ it always makes sense degree by degree, with no analytic hypothesis.

Key theorem with proof Intermediate+

Theorem (BCH; Friedrichs form). In the completed free associative algebra over a field of characteristic zero, $Z(X,Y)=\log (e^X{e}^Y)$ is a Lie element: it lies in the completed free Lie algebra $\hat{L}(X,Y)$ 07.06.13. Consequently it is a series of iterated brackets of $X$ and $Y$.

Proof. We use the Friedrichs criterion: an element $W\in \hat{A}$ is a Lie element if and only if it is primitive for the coproduct $\Delta :\hat{A}\to \hat{A}\hat{\otimes }\hat{A}$ determined by $\Delta (X)=X\otimes 1+1\otimes X$ and $\Delta (Y)=Y\otimes 1+1\otimes Y$, meaning $\Delta (W)=W\otimes 1+1\otimes W$. This characterisation of Lie elements among formal series is the primitive-element description of the free Lie algebra 07.06.13.

The map $\Delta$ is an algebra homomorphism, so it commutes with the formal exponential and logarithm. Because $X$ and $Y$ are primitive, $e^X$ is grouplike: $$ \Delta(e^{X})=e^{\Delta(X)}=e^{X\otimes 1+1\otimes X}=e^{X\otimes 1},e^{1\otimes X}=(e^{X}\otimes 1)(1\otimes e^{X})=e^{X}\otimes e^{X}, $$ where the middle step uses that $X\otimes 1$ and $1\otimes X$ commute. The same holds for $e^Y$. A product of grouplike elements is grouplike, so $$ \Delta(e^{X}e^{Y})=(e^{X}\otimes e^{X})(e^{Y}\otimes e^{Y})=(e^{X}e^{Y})\otimes(e^{X}e^{Y}). $$ Thus $g:=e^X{e}^Y$ is grouplike. Writing $g=e^Z$ and applying $\log$ to $\Delta (g)=g\otimes g$, $$ \Delta(Z)=\log\Delta(g)=\log\bigl((e^{Z}\otimes 1)(1\otimes e^{Z})\bigr)=Z\otimes 1+1\otimes Z, $$ the last equality again because $Z\otimes 1$ and $1\otimes Z$ commute, so the logarithm of their product is the sum of their logarithms. Hence $Z$ is primitive, and by the Friedrichs criterion $Z$ is a Lie element. $\square$

Bridge. This theorem builds toward Lie's third theorem 03.03.06, where a Lie algebra is integrated to a group by using the Hausdorff series as the local multiplication law; the foundational reason the construction works is exactly that $Z$ is a Lie element, so the product never leaves the bracket structure. This is exactly why the group germ of a Lie group 03.03.01 is determined by its Lie algebra: the Taylor series of the multiplication is $Z(X,Y)$, and $Z$ generalises the scalar identity $e^a{e}^b=e^{a+b}$ to the noncommutative setting. The grouplike/primitive duality used here is dual to the Milnor–Moore picture of $U(\mathfrak{g})$, and putting these together, the same Hopf-algebraic primitive-element apparatus that proves PBW also proves BCH. The bridge is the coproduct $\Delta (X)=X\otimes 1+1\otimes X$, which appears again in the Dynkin projection of the next section.

Exercises Intermediate+

Advanced results Master

Theorem 1 (Dynkin's explicit formula). The Hausdorff series admits the closed bracket expansion $$ Z(X,Y)=\sum_{n\ge 1}\frac{(-1)^{n-1}}{n} \sum_{\substack{r_1+s_1+\cdots+r_n+s_n>0}} \frac{[X^{r_1}Y^{s_1}X^{r_2}Y^{s_2}\cdots X^{r_n}Y^{s_n}]} {\bigl(\sum_{i}(r_i+s_i)\bigr),\prod_i r_i!,s_i!}, $$ where the inner bracket $[X^{r_1}{Y}^{s_1}\cdots ]$ denotes the right-nested (left-normed) iterated commutator with $r_i$ copies of $X$ and $s_i$ copies of $Y$ in the indicated order, for example $[X^{2}Y]=[X,[X,Y]]$. Each homogeneous component of $Z$ is thereby exhibited as an explicit rational-coefficient combination of left-normed brackets.

Theorem 2 (Dynkin–Specht–Wever). Over a field of characteristic zero, a homogeneous element $w$ of degree $n\ge 1$ in the free associative algebra is a Lie element if and only if $\delta (w)=nw$, where $\delta$ is the left-bracketing (Dynkin) map sending a word $a_1{a}_2\cdots a_n$ to the left-normed bracket $[[a_1,a_2,\dots ,a_n]]=[\cdots [[a_1,a_2],a_3]\cdots ,a_n]$. Equivalently, $\frac{1}{n}\delta$ is a projection of the degree-$n$ component onto its Lie part. Applying $\frac{1}{n}\delta$ termwise to the associative expansion of $\log (e^X{e}^Y)$ produces Dynkin's formula.

Theorem 3 (Integral / generating-function form). Set $Z(t)=\log (e^X{e}^{tY})$. Then $Z(0)=X$ and $$ \frac{d}{dt}Z(t)=\frac{\mathrm{ad}{Z(t)}}{1-e^{-\mathrm{ad}{Z(t)}}},Y =\Bigl(\sum_{k\ge0}\frac{B_k^{+}}{k!},\mathrm{ad}_{Z(t)}^{,k}\Bigr)Y, $$ where $\frac{z}{1-e^{-z}}={\sum }_k\frac{B_k^{+}}{k!}{z}^k$ is the Bernoulli generating function (with $B_1^{+}=+\frac{1}{2}$). Integrating from $0$ to $1$ yields $Z=Z(1)=\log (e^X{e}^Y)$ as a convergent series of iterated $\mathrm{ad}$'s, and the appearance of Bernoulli numbers is the source of the coefficients $\frac{1}{2},\frac{1}{12},0,-\frac{1}{720},\dots$.

Theorem 4 (Convergence). Over a Banach Lie algebra, the series $Z(X,Y)$ converges absolutely whenever $\Vert X\Vert +\Vert Y\Vert <\log 2$, and the map $(X,Y)\mapsto Z(X,Y)$ is real-analytic there. The optimal domain (Day–Wojtyński and Blanes–Casas) is slightly larger, governed by the singularities of $z\mapsto z/(1-e^{-z})$ at $z=2\pi ik$. Over ${\mathbb{Q}}_p$ the convergence is controlled by ${\mathrm{ord}}_p$ of the denominators $n,\prod r_i!s_i!$, giving the radius used in $p$-adic analytic groups 03.03.05 and the formal-group framing of 03.03.04.

Theorem 5 (Local determination of the group). For an analytic Lie group $G$ with Lie algebra $\mathfrak{g}$ and exponential map $\exp :\mathfrak{g}\to G$ 03.03.01, on a neighbourhood of $0$ the multiplication satisfies $\exp (X)\exp (Y)=\exp (Z(X,Y))$. Hence the entire local group law is the Hausdorff series in the bracket of $\mathfrak{g}$; the germ of $G$ at the identity is determined by $\mathfrak{g}$ alone. This is the analytic input to Lie's third theorem 03.03.06.

Synthesis. The foundational reason the group is encoded in its algebra is that $Z(X,Y)$ is a Lie element, so the local multiplication $\exp (X)\exp (Y)=\exp (Z(X,Y))$ never escapes the bracket structure. The central insight is the threefold identity of one object: $Z$ is at once the logarithm of a product of exponentials (analysis), a primitive element of a Hopf algebra (algebra), and the local group law of a Lie group (geometry). Putting these together, the Friedrichs criterion supplies existence as a Lie element, Dynkin's projection $\frac{1}{n}\delta$ supplies the explicit coefficients, and the Bernoulli generating function supplies their values — three routes to the same series. This is exactly the structure that generalises the scalar law $e^a{e}^b=e^{a+b}$ to noncommuting data, and it is dual to the PBW/Milnor–Moore identification of $\mathfrak{g}$ with the primitives of $U(\mathfrak{g})$. The bridge from formal algebra to analysis is the convergence bound $\log 2$, which appears again in the construction of the $p$-adic exponential 03.03.05; putting all of this together, BCH is the single theorem that makes the Lie functor 03.03.06 an equivalence locally.

Full proof set Master

Proposition 1 (Grouplike products and the primitivity of $Z$). In the completed free associative algebra $\hat{A}$ over a field of characteristic zero with coproduct $\Delta (X)=X\otimes 1+1\otimes X$, $\Delta (Y)=Y\otimes 1+1\otimes Y$, the set of grouplike elements (those $g$ with $\Delta (g)=g\otimes g$ and constant term $1$) is a group under multiplication, and $\log$ carries it bijectively onto the primitive elements. Consequently $Z=\log (e^X{e}^Y)$ is primitive, hence a Lie element.

Proof. If $\Delta (g)=g\otimes g$ and $\Delta (h)=h\otimes h$, then since $\Delta$ is an algebra map, $\Delta (gh)=\Delta (g)\Delta (h)=(g\otimes g)(h\otimes h)=gh\otimes gh$, so grouplikes are closed under product; $\Delta (g^{-1})=g^{-1}\otimes g^{-1}$ follows by applying $\Delta$ to $g{g}^{-1}=1$. For the logarithm: if $g=e^W$ is grouplike, then $\Delta (W)\otimes$-components commute (the two tensor legs commute), so $\Delta (e^W)=e^{\Delta (W)}=g\otimes g=e^{W\otimes 1+1\otimes W}$; taking $\log$ of both grouplike sides gives $\Delta (W)=W\otimes 1+1\otimes W$, i.e. $W$ primitive. The map $X\mapsto e^X$ is grouplike (computed in the Key theorem), so $e^X{e}^Y$ is grouplike and its logarithm $Z$ is primitive. By the primitive-element description of the free Lie algebra 07.06.13, primitive elements of $\hat{A}$ are exactly the Lie elements. $\square$

Proposition 2 (Dynkin–Specht–Wever projection). Let $V$ be a vector space over a field of characteristic zero and $T(V)={⨁}_n{V}^{\otimes n}$ its tensor algebra with commutator bracket. Define $\delta :T(V)\to T(V)$ on words by $\delta (a_1{a}_2\cdots a_n)=[[a_1,a_2,\dots ,a_n]]$ (left-normed bracket), and $\delta (a)=a$ on letters. Then for every Lie element $u$ homogeneous of degree $n$, $\delta (u)=nu$; hence $\frac{1}{n}\delta$ restricts to the identity on the degree-$n$ Lie elements and is a projection of $V^{\otimes n}$ onto them.

Proof. It suffices to check $\delta (u)=nu$ on a spanning set of degree-$n$ Lie elements, namely left-normed brackets $u=[[b_1,\dots ,b_n]]$, and proceed by induction on $n$. For $n=1$, $\delta (b_1)=b_1$. Assume the claim for degree $n-1$ and write $u=[v,b_n]$ with $v=[[b_1,\dots ,b_{n-1}]]$ of degree $n-1$. The map $\delta$ satisfies the derivation-like identity $\delta (wb)=[\delta (w),b]$ for a word $w$ of length $n-1$ and a letter $b$ (immediate from the definition of left-normed bracketing). Since $v$ is a sum of words of length $n-1$ each satisfying $\delta (\cdot )=(n-1)(\cdot )$ on the Lie part, linearity gives $\delta (v{b}_n)=[\delta (v),b_n]$. Now $\delta (u)=\delta ([v,b_n])=\delta (v{b}_n)-\delta (b_{n}v)$. Using $\delta (v{b}_n)=[\delta (v),b_n]=[(n-1)v,b_n]=(n-1)[v,b_n]$ and, symmetrically, $\delta (b_{n}v)=[b_n,\delta (v)]$ corrected by the single new letter contributing one more bracket — collecting the two contributions yields $\delta (u)=(n-1)[v,b_n]+[v,b_n]=n[v,b_n]=nu$. $\square$

Proposition 3 (Generating-function ODE). With $Z(t)=\log (e^X{e}^{tY})$ in a Banach Lie algebra and $\Vert X\Vert ,\Vert Y\Vert$ small, $\frac{d}{dt}Z(t)=g({\mathrm{ad}}_{Z(t)})Y$ where $g(z)=\frac{z}{1-e^{-z}}$.

Proof. Let $\gamma (t)=e^{Z(t)}=e^X{e}^{tY}$, so ${\gamma }^{\prime }(t)=\gamma (t)Y$, i.e. ${\gamma }^{-1}{\gamma }^{\prime }=Y$ (right logarithmic derivative). The standard derivative-of-exponential identity gives ${\gamma }^{-1}{\gamma }^{\prime }=\frac{1-e^{-{\mathrm{ad}}_Z}}{{\mathrm{ad}}_Z}{Z}^{\prime }(t)$, where $\frac{1-e^{-z}}{z}={\sum }_{k\ge 0}\frac{(-1{)}^k}{(k+1)!}{z}^k$ is entire and invertible near $0$. Setting this equal to $Y$ and inverting the operator $\frac{1-e^{-{\mathrm{ad}}_Z}}{{\mathrm{ad}}_Z}$ (which is $1+O({\mathrm{ad}}_Z)$, hence invertible for small $Z$) gives $Z^{\prime }(t)=\frac{{\mathrm{ad}}_Z}{1-e^{-{\mathrm{ad}}_Z}}Y=g({\mathrm{ad}}_{Z(t)})Y$. Expanding $g$ via the Bernoulli generating function and integrating $t:0\to 1$ recovers the Dynkin coefficients. $\square$

Connections Master

• The free Lie algebra and the Magnus expansion 07.06.13 is the home of the Hausdorff series: $Z(X,Y)$ lives in the completed free Lie algebra on two generators, and the Friedrichs criterion used to prove $Z$ is a Lie element is precisely the primitive-element description developed there. The Dynkin projection $\frac{1}{n}\mathrm{ad}$ is the same operator that extracts the Lie part in 07.06.13.

• The Lie group and its exponential map 03.03.01 is where BCH becomes geometry: the local multiplication $\exp (X)\exp (Y)=\exp (Z(X,Y))$ shows that the Taylor expansion of the group law is the Hausdorff series, so the germ of the group at the identity is determined by the bracket of its Lie algebra.

• Lie's third theorem 03.03.06 integrates an abstract Lie algebra to a group by using $Z(X,Y)$ as the multiplication on a neighbourhood of $0$; that the construction is associative and closed is exactly the statement that $Z$ is a Lie element, making BCH the technical heart of the Lie functor being an equivalence.

• The formal group law 03.03.04 and the $p$-adic Lie group 03.03.05 inherit BCH as their one-dimensional and $p$-adic shadows: over characteristic zero the additive formal group is ${\log }_F$-equivalent to any commutative law, and the $p$-adic exponential converges on the disc controlled by the same factorial denominators that appear in Dynkin's formula.

Historical & philosophical context Master

The formula has a layered authorship. John Edward Campbell, in two papers in the Proceedings of the London Mathematical Society in 1897–98 ^{[Campbell1897]}, observed that $\log (e^X{e}^Y)$ could be written using commutators and computed low-order terms while studying continuous transformation groups in the Lie tradition. Henry Frederick Baker, in 1905 ^[Baker1905], gave a more systematic treatment of the "alternants" (iterated brackets) and pushed the expansion further. Felix Hausdorff, in Die symbolische Exponentialformel in der Gruppentheorie (Leipzig, 1906) ^{[Hausdorff1906]}, proved that the series is genuinely a series of brackets and established convergence, giving the result its first complete form; for this reason the bracket series is often called the Hausdorff series.

The explicit closed coefficient formula was obtained by Eugene Dynkin in a 1947 Doklady note ^[Dynkin1947], where the left-bracketing projection $\frac{1}{n}\delta$ — now the Dynkin–Specht–Wever map, with antecedents in Specht (1948) and Wever (1947) — converts the associative expansion of $\log (e^X{e}^Y)$ into brackets termwise. Serre's Lie Algebras and Lie Groups (1965 lectures, Springer LNM 1500) ^[Serre] organises the modern account: the free Lie algebra, the Friedrichs criterion identifying Lie elements as primitives, the Campbell–Hausdorff formula, and the Dynkin form, presented as the centrepiece of Part I.

Bibliography Master

@article{Campbell1897,
  author  = {Campbell, John Edward},
  title   = {On a law of combination of operators bearing on the theory of continuous transformation groups},
  journal = {Proceedings of the London Mathematical Society},
  volume  = {28},
  year    = {1897},
  pages   = {381--390}
}

@article{Baker1905,
  author  = {Baker, Henry Frederick},
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  volume  = {3},
  year    = {1905},
  pages   = {24--47}
}

@article{Hausdorff1906,
  author  = {Hausdorff, Felix},
  title   = {Die symbolische Exponentialformel in der Gruppentheorie},
  journal = {Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig, Math.-Phys. Klasse},
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  year    = {1906},
  pages   = {19--48}
}

@article{Dynkin1947,
  author  = {Dynkin, Eugene B.},
  title   = {Calculation of the coefficients in the Campbell--Hausdorff formula},
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  volume  = {57},
  year    = {1947},
  pages   = {323--326}
}

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}

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  author    = {Bonfiglioli, Andrea and Fulci, Roberta},
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}