a) Show that for all $x, y, z, t \in \mathbb{R}$ we have
$$N(xE + yI + zJ + tK) = x^2 + y^2 + z^2 + t^2.$$
b) Show that for all $U \in \mathbb{H}^{\mathrm{im}}$ we have $U^2 = -N(U)E$ and that
$$\mathbb{H}^{\mathrm{im}} = \left\{ U \in \mathbb{H} \mid U^2 \in \left]-\infty, 0\right] E \right\}.$$