= Solution
Use $JE_1=-E_2$, $E_1J=E_2$ and $JE_1J=E_1$ to expand the <matrix product>:
$$
AE_1A^T=c^2E_1+cs(JE_1-E_1J)-s^2JE_1J=(c^2-s^2)E_1-2csE_2.
$$
The double-angle identities therefore give
$$
\boxed{AE_1A^T=\cos\theta\,E_1-\sin\theta\,E_2.}
$$
Similarly $JE_2=E_1$, $E_2J=-E_1$ and $JE_2J=E_2$, so
$$
\boxed{AE_2A^T=\sin\theta\,E_1+\cos\theta\,E_2.}
$$
This <conjugation> rotates the two-dimensional span of the two matrices through the double angle described by <orthogonal rotations generated by anticommuting involutions>.
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