Unfortunately, the answer is 'no'. You are basically asking whether you can simultaneously diagonalize two quadratic forms in three variables, and the answer is that, 'generically' you can (and you always can if some linear combination of the two is definite), but there are special pairs that cannot be simultaneously diagonalized.

This happens already in dimension $2$. You can't simultaneously diagonalize $x^2$ and $xy$, for example. I think you cannot simultaneously diagonalize $-x^2 + y^2 + z^2$ and $2xy + z^2$ (if I remember the example correctly).

Generally, if the (real) null cones of the two indefinite quadratic forms are tangent, then they can't be simultaneously diagonalized.