Quaternion ($\mathbb{H}$)

A quaternion is $a+bi+cj+dk$ with real coefficients, where $i^2=j^2=k^2=ijk=-1$. They form a noncommutative real division algebra. The norm squared is $a^2+b^2+c^2+d^2$, and every nonzero quaternion has inverse its conjugate divided by that squared norm. The unit quaternions form the Lie group $S^3$.