Let $f$ be an automorphism of the algebra of octonions. Is it true that $f$ preserves some quaternionic subalgebra? Has the statement an elementary proof?



About some property of automorphism of octonions
Seen as a map of $8$-dimensional Euclidean vector spaces, $f$ is obviously (special) orthogonal, so we can find an orthonormal basis on which is has a block diagonal form of $2\times 2$ rotation matrices, and certainly at least $u,v$ unit octonions, orthogonal to each other and both orthogonal to $1$ (=pure imaginary) such that $f(u) = \cos\theta\cdot u + \sin\theta\cdot v$ and $f(v) = -\sin\theta\cdot u + \cos\theta\cdot v$. Then $1,u,v,uv$ span (and in fact, are an orthonormal basis of) a quaternion algebra which is preserved by $f$ (incidentally, the unit imaginary octonion $uv$ is also preserved by $f$).


