Someone please shed some light to my confused brain: "The statement says that a 4 by 4 real matrix B has 4 eigenvalues $\alpha, \bar{\alpha}, -\alpha, -\bar{\alpha}$ where $\alpha = a+ib$, $a,b$ both not zero. Then it is obvious that the Jordan form of B is diagonal."
My question: Does this mean diagonal in the literal sense? Or should it look like J1 + J2, where
J1(1,1)= a, J1(1,2)= b, J1(2,1)=-b, J1(2,2)=a
J2(1,1)=-a, J2(1,2)=-b, J2(2,1)=-a, J2(2,2)=b
$+$ means direct sum.