This is really a simple question. Let $A$ be an associative, nilpotent real algebra, and set $[a,b]=ab-ba$, define the exponential map as usual, that is $exp(a)=1+a+\frac{a^2}{2}+...$.
Let $G=exp(A)$, and consider $L:G \rightarrow GL(G)$ such that $L(g)=L_g$, the left translation. One can consider its derivative at the identity, $L^*:G \rightarrow GL(A)$.
My question is, how do I do I take the derivative of $L^*$?