For $(a, b) \in \mathbb{C}^2$, we denote by $M(a, b)$ the square complex matrix $M(a, b) = \left( \begin{array}{cc} a & -b \\ \bar{b} & \bar{a} \end{array} \right) \in \mathcal{M}_2(\mathbb{C})$. Every element $q \in \mathbb{H}$ can be written uniquely as $q = xe + yI + zJ + tK$ with $x, y, z, t \in \mathbb{R}$. For $x, y, z, t \in \mathbb{R}$ and $q = xe + yI + zJ + tK \in \mathbb{H}$ we set $q^* = xe - yI - zJ - tK \in \mathbb{H}$ and $N(q) = x^2 + y^2 + z^2 + t^2 \in \mathbb{R}_+$. a) Let $(x, y, z, t) \in \mathbb{R}^4$ and $q = xe + yI + zJ + tK$. Express the trace of the matrix $q \in \mathcal{M}_2(\mathbb{C})$ in terms of the real number $x$. b) Deduce that, for all $(q, r) \in \mathbb{H}^2$, $qr - rq = q^* r^* - r^* q^*$. c) Let $a, b, c, d$ be quaternions. Establish the relation $(acb^*)d + d^*(acb^*)^* = (acb^*)^* d^* + d(acb^*)$. Deduce the identity $(N(a) + N(b))(N(c) + N(d)) = N(ac - d^* b) + N(bc^* + da)$.
For $(a, b) \in \mathbb{C}^2$, we denote by $M(a, b)$ the square complex matrix $M(a, b) = \left( \begin{array}{cc} a & -b \\ \bar{b} & \bar{a} \end{array} \right) \in \mathcal{M}_2(\mathbb{C})$. Every element $q \in \mathbb{H}$ can be written uniquely as $q = xe + yI + zJ + tK$ with $x, y, z, t \in \mathbb{R}$. For $x, y, z, t \in \mathbb{R}$ and $q = xe + yI + zJ + tK \in \mathbb{H}$ we set $q^* = xe - yI - zJ - tK \in \mathbb{H}$ and $N(q) = x^2 + y^2 + z^2 + t^2 \in \mathbb{R}_+$.\\
a) Let $(x, y, z, t) \in \mathbb{R}^4$ and $q = xe + yI + zJ + tK$. Express the trace of the matrix $q \in \mathcal{M}_2(\mathbb{C})$ in terms of the real number $x$.\\
b) Deduce that, for all $(q, r) \in \mathbb{H}^2$, $qr - rq = q^* r^* - r^* q^*$.\\
c) Let $a, b, c, d$ be quaternions. Establish the relation $(acb^*)d + d^*(acb^*)^* = (acb^*)^* d^* + d(acb^*)$.
Deduce the identity $(N(a) + N(b))(N(c) + N(d)) = N(ac - d^* b) + N(bc^* + da)$.