The _Baker-Campbell-Hausdorff lemma_ or _BCH lemma_ may be defined as follows or in terms of an [integral equation.](Baker-Campbell-Hausdorff%20lemma.md#Integral%20form%20of%20the%20BCH%20lemma)
Given the [adjoint map](Adjoint%20map.md) $\mbox{Ad}_{e^X}(Y) = e^{ad_X}(Y)=e^XYe^{-X}$ where $X$ and $Y$ belong in a [complex vector space](Complex%20vector%20spaces.md),
$e^XYe^{-X}=Y+[X,Y]+\frac{1}{2!}[X,[X,Y]]+...+\frac{1}{n!}[X,[X,[X,...[X,Y]]]...]+...$
where the $[X,Y]$ terms are [commutators.](Commutators.md)
This is the derivative form of the BCH lemma since the adjoint map may be written as a derivative such that [$\mbox{Ad}_X(Y) = \frac{d}{dt}e^{tX}Ye^{-tX}\bigg|_{t=0} = [X,Y].$](Proof%20that%20the%20commutator%20is%20an%20adjoint%20map%20of%20a%20Lie%20Algebra#^0ac989)
%%I don't get this last statement%%
# Complex BCH lemma
The lemma is identical for [complex matrix exponentials](Complex%20matrix%20exponentials.md), and thus we may write: $e^{iX\lambda}Ye^{-iX\lambda}=Y+i\lambda[X,Y]+\frac{i^2\lambda^2}{2!}[X,[X,Y]]+...+\frac{i^n\lambda^n}{n!}[X,[X,[X,...[X,Y]]]...]+...$ where $\lambda \in \mathbb{R}$.
%%Does X or do X and Y also have to be Hermitian? Is there any connection to unitarity?%%
# Integral form of the BCH lemma
We find that adjoint map, $e^XYe^{-X}$ is also a solution to the 1st order integral equation $Y(t) = Y + \int_0^t d\tau [X,Y(\tau)].$ ^5f9ea7
This integral can be expanded to any number of $n$th order equations $Y(t) = Y + \int_0^{t} d\tau_1 [X,Y(\tau_1)]+ \frac{t}{2!}\int_0^{t}\int_0^{t} d\tau_1d\tau_2[X,[X,Y(\tau_2)]]+...$
$+\frac{t^{n-1}}{n!}\int_0^{t}...\int_0^{t} d\tau_1...d\tau_n [X,[X,[X,...[X,Y(\tau_n)]]]...]$ thus giving an integral form of the BCH lemma.
%%We need to define integral equations. We won't link this to an empty note referred to as an "integral equation" yet simply because integral equations aren't defined or this may as well be equivalent to an integral form of a 1st order ODE.%%
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# Proofs and Examples
## Proof of the Baker-Campbell-Hausdorff lemma
### Proof of the integral form of the BCH lemma
#MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory/Lie/LieGroups
#MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory/Lie/LieGroups/Algebras/LieAlgebras
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