Algebraic Structure and Abstract Properties of Complex Numbers
The question asks to prove structural properties (e.g., subgroup, vector space, division ring) or abstract algebraic results about sets of complex numbers or generalizations like quaternions.
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 $$\mathbb{H}^{\mathrm{im}} = \left\{ U \in \mathbb{H} \mid U^2 \in \left]-\infty, 0\right] E \right\}.$$