 Orthogonal rotations generated by anticommuting involutions

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 $R(t)=e^{tJ}=\cos tI+\sin tJ$. Multiplication gives $R(t{)}^{T}R(t)=I$ and the conjugation identitiesIndeed $JE=-F$, $EJ=F$, $JF=E$, 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.