  For an arbitrary field $\mathbb{K}$ and a family of inner products in a
$\mathbb{K}$-vector space $V$ of arbitrary dimension, we study necessary and
sufficient conditions in order to have an orthogonal basis relative to all the
inner products. If the family contains a nondegenerate element plus a
compatibility condition, then under mild hypotheses the simultaneous
orthogonalization can be achieved. So we investigate several constructions
whose purpose is to add a nondegenerate element to a degenerate family and we
study under what conditions the enlarged family is nondegenerate.
