  We explain in this note how real fermionic and bosonic quadratic forms can be
effectively diagonalized. Nothing like that exists for the general complex
hermitian forms. Looks like this observation was missed in the Quantum Field
theoretical literature. The present author observed it for the case of fermions
in 1986 making some topological work dedicated to the problem: how to construct
Morse-type inequalities for the generic real vector fields? This idea also is
presented in the note.
