Existence of flipped orthogonal conjugate symmetric Jordan canonical bases for real H-selfadjoint matrices

For real matrices selfadjoint in an indefinite inner product there are two special canonical Jordan forms, that is (i) flipped orthogonal (FO) and (ii) $\gamma$-conjugate symmetric (CS). These are the classical Jordan forms with certain additional properties induced by the fact that they are $H$-selfadjoint. In this paper we prove that for any real $H$-selfadjoint matrix there is a $\gamma$-FOCS Jordan form that is simultaneously flipped orthogonal and $\gamma$-conjugate symmetric.