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 $\mathbf{H} = \{I_3 + M \mid M \in \mathbf{L}\}$, with the group law $M * N = M + N + \frac{1}{2}[M,N]$ on $\mathbf{L}$. Show that $\mathbf{H}$ equipped with the usual product of matrices is a subgroup of $\mathrm{SL}_3(\mathbf{R})$ and that $$\exp : (\mathbf{L}, *) \rightarrow (\mathbf{H}, \times)$$ is a group isomorphism.
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 $\mathbf{H} = \{I_3 + M \mid M \in \mathbf{L}\}$, with the group law $M * N = M + N + \frac{1}{2}[M,N]$ on $\mathbf{L}$.
Show that $\mathbf{H}$ equipped with the usual product of matrices is a subgroup of $\mathrm{SL}_3(\mathbf{R})$ and that
$$\exp : (\mathbf{L}, *) \rightarrow (\mathbf{H}, \times)$$
is a group isomorphism.