Let $U, V \in \mathbb{H}^{\mathrm{im}}$.\\
a) Show that $U$ and $V$ are orthogonal if and only if $UV + VU = 0$. In this case show that $UV \in \mathbb{H}^{\mathrm{im}}$ and that the determinant of the family $(U, V, UV)$ in the basis $(I, J, K)$ of $\mathbb{H}^{\mathrm{im}}$ is non-negative.\\
b) Show that if $(U, V)$ is an orthonormal family in $\mathbb{H}^{\mathrm{im}}$, then $(U, V, UV)$ is a direct orthonormal basis of $\mathbb{H}^{\mathrm{im}}$.