We consider the set of square matrices of size 3 that are strictly upper triangular:
$$\mathbf{L} = \left\{ M_{p,q,r} \mid (p,q,r) \in \mathbf{R}^3 \right\} \quad \text{where} \quad M_{p,q,r} = \begin{pmatrix} 0 & p & r \\ 0 & 0 & q \\ 0 & 0 & 0 \end{pmatrix},$$
and the group law $M * N = M + N + \frac{1}{2}[M,N]$ on $\mathbf{L}$.
Show that for all matrices $M, N \in \mathbf{L}$, we have:
$$(\exp M) \times (\exp N) = \exp(M * N)$$