@hallasurvivor I vaguely remembered something about all unit vectors being roots of -1 in the quaternions, and I only just remembered a fun way to see it, which is that _every_ quaterion q is a root of a quadratic polynomial over 𝐑, namely χ=X²-tr(q)X+det(q), where
tr(q)=2Re(q)=2a
is twice the real part of q=a+ib+jc+kd and
det(q)=a²+b²+c²+d²
is the square of the absolute value of q (χ is the characteristic polynomial à la
https://arxiv.org/abs/math/0203276) so taking a=0 gives you an S² worth of roots. 𝐑 is the center of 𝐇, so the map 𝐑[X]→𝐇 that evaluates in q is a ring homomorphism, and its kernel (the ideal of polynomials with q as a root) is generated by χ for q∉𝐑, which is a (roundabout) way of seeing that that only the q∉𝐑 with Re(q)=0 and |q|=1 make the cut. Incidentally χ factors over the (commutative) subring 𝐑[q]≅𝐂 generated by 𝐑 and q as χ=(X-q)(X-q*), since q*=2Re(q)-q is in there and we have tr(q)=q+q* and det(q)=qq*=q*q.
