We consider two matrices $A$ and $B$ of $\mathcal{M}_d(\mathbf{R})$. We further assume that $A$ and $B$ commute with $[A,B]$.
(a) Show that $[A, \exp(B)] = \exp(B)[A,B]$.
(b) Determine a differential equation satisfied by $t \mapsto \exp(tA)\exp(tB)$.
(c) Deduce the formula:
$$\exp(A)\exp(B) = \exp\left(A + B + \frac{1}{2}[A,B]\right)$$