Matrix Lie groups

Let $H$ be the Heisenberg group of real matrices $\begin{pmatrix}1&a&c\\0&1&b\\0&0&1\end{pmatrix}$. (a) Show that its Lie algebra $\mathfrak h$ is the space of strictly upper-triangular real $3\times3$ matrices, and compute the brackets of $P = E_{12}$, $Q = E_{23}$, $Z = E_{13}$. (b) Compute $e^X$ for $X = \begin{pmatrix}0&x&z\\0&0&y\\0&0&0\end{pmatrix}$ and deduce that $\exp : \mathfrak h\to H$ is a bijection. (c) Find the centre of $H$ and of $\mathfrak h$.