Stuff which squares to $-1$ in the quaternions, thinking geometrically.

Question

How can we think of the set$$\{x \in \mathbb{H} : x^2 = -1\}$$geometrically? Is this set finite or infinite? Are there some more geometric ways of thinking about than meets the eye? Here, $\mathbb{H}$ denotes the quaternions.

Answer 1

First write $x=a+bi+cj+dk$ and compute that $$x^2=a^2+2abi+2acj+2adk-b^2-c^2-d^2.$$ This yields the equations $$\begin{align}a^2-b^2-c^2-d^2&=-1\\ab&=0\\ac&=0\\ad&=0.\end{align}$$ If $a\neq 0$ then $b=c=d=0$ and $a^2=-1$, which is impossible. Hence, $a=0$ and $b^2+c^2+d^2=1$. Hence, there are infinitely many elements $x\in\mathbb{H}$ satisfying $x^2=-1$. This set can be characterized as the unit sphere in the hyperplane spanned by $i,j,k$.