For $(a, b) \in \mathbb{C}^2$, we denote by $M(a, b)$ the square complex matrix $M(a, b) = \left( \begin{array}{cc} a & -b \\ \bar{b} & \bar{a} \end{array} \right) \in \mathcal{M}_2(\mathbb{C})$. A matrix of the form $M(a, b)$ will be called a quaternion. We will consider in particular the quaternions $e = I_2 = M(1, 0)$, $I = M(0, 1)$, $J = M(\mathrm{i}, 0)$, $K = M(0, -\mathrm{i})$ and we will denote by $\mathbb{H} = \{M(a, b) \mid (a, b) \in \mathbb{C}^2\}$ the subset of $\mathcal{M}_2(\mathbb{C})$ consisting of all quaternions. a) Calculate the products pairwise of the matrices $e, I, J, K$. Present the results in a double-entry table. b) Deduce that $(\mathrm{i} I, \mathrm{i} J, \mathrm{i} K)$ is an H-system.
For $(a, b) \in \mathbb{C}^2$, we denote by $M(a, b)$ the square complex matrix $M(a, b) = \left( \begin{array}{cc} a & -b \\ \bar{b} & \bar{a} \end{array} \right) \in \mathcal{M}_2(\mathbb{C})$. A matrix of the form $M(a, b)$ will be called a quaternion. We will consider in particular the quaternions $e = I_2 = M(1, 0)$, $I = M(0, 1)$, $J = M(\mathrm{i}, 0)$, $K = M(0, -\mathrm{i})$ and we will denote by $\mathbb{H} = \{M(a, b) \mid (a, b) \in \mathbb{C}^2\}$ the subset of $\mathcal{M}_2(\mathbb{C})$ consisting of all quaternions.\\
a) Calculate the products pairwise of the matrices $e, I, J, K$. Present the results in a double-entry table.\\
b) Deduce that $(\mathrm{i} I, \mathrm{i} J, \mathrm{i} K)$ is an H-system.