In joint work with Andreas Fischle (TU Dresden, Germany) and Patrizio Neff (U Essen, Germany) we needed to use the following statement: If $S$ is a real $n\times n$ matrix with $S^2$ symmetric, then there exists an orthogonal matrix $P$ such that $PSP^{-1}$ is block diagonal with blocks of size at most two. While we have a proof of this statement, we are wondering whether it already appears in the literature. Have you seen this result or its generalizations published anywhere?

reasonablecite-able source in contemporary mathematics. $\endgroup$9more comments