Number of solutions of the quaternion

Question

I have the quaternions $q = x_0 + ix_i + jx_2 + kx_3$ where $x_i \in \mathbb{R}$ and $i,j,k$ satisfy the relations $$ i^2=j^2=k^2 = -1, \hspace{0.3cm} ij=-ji=k,\hspace{0.3cm} jk=-kj=i,\hspace{0.3cm} ki=-ik=j$$ I also have that $f(q) =q^2$ defines a smooth map from $\mathbb{R}^4 \cup \{\infty\} \cong S^4$ to itself.

I have to find how many solutions are there to the equation $q^2 = 1$.

Can someone help me ?

Answer 1

Write the quaternion $q$ as $x_0 + v$ where $v = i x_1 + j x_2 + k x_3$, and note that $q^2 = x_0^2 + 2 x_0 v - (x_1^2 + x_2^2 + x_3^2)$. Thus $q^2 = 1$ implies $x_0 v = 0$, i.e. either $x_0 = 0$ or $v=0$. With $x_0 = 0$ we would have $x_1^2 + x_2^2 + x_3^2 = -1$, which has no real solutions. With $v=0$ we have $x_0 = \pm 1$.