Simultaneous decomplexification of a pair of complex matrices by a similarity transformation

The following question is treated: Under what conditions can complex n-by-n matrices A and B be made real by the same similarity transformation? It is shown that if the algebra generated by A and B contains a matrix with a simple real spectrum, then the problem of the simultaneous decomplexification of a matrix pair can be reduced to the decomplexification of a single matrix by a diagonal similarity transformation. From this result, sufficient conditions are derived for the possibility of simultaneous decomplexification. An example illustrating these conditions is given.