Scalar square of a traceless two-by-two matrix
= Scalar square of a traceless two-by-two matrix
{title2=$M^2=-(\det M)I$}
A two-by-two <matrix> $M$ with zero <trace> satisfies $M^2=-(\det M)I$ by the <Cayley-Hamilton theorem>. Over the <complex numbers>, either $\det M\ne0$ and its two distinct <eigenvalues> make it diagonalizable, or $M^2=0$. A matrix commutator has zero trace, so this alternative applies to it. Diagonalization over the real numbers need not follow when $-\det M<0$.