Command for real Jordan decomposition?

Hello,

I would like to perform a real Jordan decomposition of a real square matrix. That is, I am looking for the Jordan decomposition where any pair of complex conjugate eigenvalues yields a real Jordan block [[a,b];[a,-b]] instead of the complex block [[a + ib, *]; [0, a - ib]]. (A detailed discussion on this can be found in the book of Hirsch and Smale.)

Since jor(A) provides one with a complex decomposition whenever A has at least one pair of complex eigenvalues, my hope was that cdf2rdf would provide me with what I am looking for. Unfortunately, this is not the case, as I could establish on an example.

Did I miss a command or may it be really the case that one has to derive the real Jordan decomposition from the complex one using its own code? It would be stunning if MATLAB would not provide such a command ...