= Orthogonal rotations generated by anticommuting involutions
{title2=$R(t)=\cos t\,I+\sin t\,EF$}
Suppose real <symmetric matrices> $E,F$ satisfy $E^2=F^2=I$ and $EF=-FE$. The <involutions> are <orthogonal matrices>, and $J=EF$ satisfies $J^T=-J$, $J^2=-I$. The <matrix exponential> therefore reduces to $R(t)=e^{tJ}=\cos t\,I+\sin t\,J$. Multiplication gives $R(t)^TR(t)=I$ and the <conjugation> identities
$$
R(t)ER(t)^T=\cos(2t)E-\sin(2t)F,\qquad R(t)FR(t)^T=\sin(2t)E+\cos(2t)F.
$$
Indeed $JE=-F$, $EJ=F$, $JF=E$, and $FJ=-E$, which give both identities by expansion. The angle in this two-dimensional space of matrices is twice the parameter in the orthogonal matrix. These formulas hold in any finite dimension where the hypotheses can be satisfied.
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