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 $S^3$.

The displayed complex matrices, with real coefficients, form a four-dimensional real associative algebra. Their three imaginary basis matrices square to minus the identity and satisfy $JK=L$, $KL=J$, $LJ=K$, with the reversed products negative. For $\alpha =aI+bJ+cK+dL$, quaternionic conjugation gives $\alpha \overline{\alpha }=(a^2+b^2+c^2+d^2)I$. Every nonzero matrix therefore has inverse $\overline{\alpha }/(a^2+b^2+c^2+d^2)$ inside this algebra. This realizes the real division algebra of quaternions within $M_2(\mathbb{C})$; it is not a complex vector subspace.

A quaternion of norm $1$ is a unit quaternion. They form the topological group $\mathrm{Sp}(1)$ and identify with the sphere $S^3\subset {\mathbb{R}}^4$. Multiplication and inversion preserve the norm. The Hurewicz theorem identifies ${\pi }_3(S^3)$ with $H_3(S^3;\mathbb{Z})$; hence homotopy-group addition in a topological group shows that $q\mapsto q^m$ has mapping degree $m$, rather than $m^2$.