Real symmetric generalized eigenvalue problem

For the generalized eigenvalue problem Ax=eBx, where A and B are real
symmetric square matrices, x is the eigenvector, and e is the eigenvalue,
the eigenvalues are supposed to be real. Are there other requirements for e
to be real besides A and B's being real symmetric? For example, if A=[1,0;
0,-1] and B=[0,1; 1,0], e=+-i. However, B^{-1}A is skew-symmetric, and
skew-symmetric matrices have imaginary eigenvalues. I do not recall ever
seeing (and cannot now find) any requirements on B^{-1}A in order for the
generalized eigenvalue problem to have real eigenvalues. What am I missing?


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