Question: I was reading Lie algebra text. In that, I came across the following.

If $A$ and $B \in \mathbb{C}^{d \times d},$ and $A B=B A,$ then
$$
\exp (A) \exp (B)=\exp (B) \exp (A)=\exp (A+B).
$$
But this fails when $A B \neq B A .$

Still we have, $$
\lim \_{k \rightarrow \infty}(\exp (A / k) \exp (B / k))^{k}=\exp (A+B) \quad\left(A, B \in \mathbb{C}^{d \times d}\right).
$$

I am interested in looking for any application of the above fact. I have gone through Gerd Herzog's proof, which is very elementary. But I could not get any applications. If anyone of you encountered it, please share it here.

Thank you very much.

Answer: 
here are a couple of applications I like.

**1.)** The Lie Product Formula makes it obvious that the infinitessimal generator for a markov process behaves like a standard markov matrix under the image of the exponential map. I.e. $G:= Q+D$ with $Q$ having real non-negative entries (and only zeros on the diagonal) and $D$ being a diagonal matrix with only negative entries on the diagonal such that $G\mathbf 1=\mathbf 0$. This implies $\exp\big(G\big)\mathbf 1 = \mathbf 1$ and  

$\exp\big(G\big)=\lim \_{k \rightarrow \infty}\Big(\exp \big(\frac{Q}{k}\big) \exp \big(\frac{D}{k}\big)\Big)^{k}$  

where for all natural numbers $k$ we see $\exp\big(\frac{Q}{k}\big)$ and $ \exp \big(\frac{D}{k}\big)$ are real non-negative, hence their product is and so is the limit $=\exp\big(G\big)$. Thus $\exp\big(G\big)$ is a stochasitic matrix.

**2.)** The Golden-Thompson Inequality, that for Hermitian $A$ and $B$  

$\text{trace}\Big(e^{A+B}\Big)\leq \text{trace}\Big(e^Ae^B\Big)$
with equality *iff* $AB = BA$.

There are many proofs but the nicest ones use Lie Product formula in my view. In particular I like the proof by Vershynin  

[https://www.math.uci.edu/~rvershyn/papers/golden-thompson.pdf](https://www.math.uci.edu/%7Ervershyn/papers/golden-thompson.pdf)
I'm not sure if this is the kind of thing you're looking for, but here are two examples where the result can be applied.

* Proving that the Lie algebra of a matrix group is closed under addition (where $X$ is defined to be an element of the Lie algebra of $G$ iff $e^{tX} \in G$ for all $t \in \Bbb R$)
* Proving that a homomorphism $\Phi : G \to H$ of matrix groups induces a Lie algebra homomorphism $\phi:\mathfrak g \to \mathfrak h$ between the associated Lie algebras