  We show that under certain conditions every maximal symmetric subfield of a
central division algebra with positive unitary involution $(B,\tau)$ will be a
Galois extension of the fixed field of $\tau$ and will "real split" $(B,\tau)$.
As an application we show that a sufficient condition for the existence of
positive involutions on certain crossed product division algebras, considered
by Berhuy in the context of unitary space-time coding, is also necessary,
proving that Berhuy's construction is optimal.
