Let $\Gamma(\mathbb{K})$ be the subset of $\mathcal{M}_{2}(\mathbb{K})$ consisting of matrices of the form $\left( \begin{array}{cc} a & -b \\ b & a \end{array} \right)$ where $(a, b) \in \mathbb{K}^{2}$.
Show that $\left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right)$ is diagonalisable over $\mathbb{C}$. Deduce that $\Gamma(\mathbb{C})$ is a diagonalisable subalgebra of $\mathcal{M}_{2}(\mathbb{C})$.