Proposition: No equivariant signature change from an inner product alone

For $n\ge 2$, no rule assigns to each positive-definite inner product on an $n$-dimensional real vector space a Lorentzian bilinear form on that same space, using only that inner product and equivariantly under all linear changes of coordinates.

Proof. At the Euclidean inner product $I$, equivariance makes the assigned form invariant under every orthogonal transformation. Coordinate reflections force its off-diagonal entries to vanish. Coordinate permutations force its diagonal entries to coincide. It is therefore $\lambda I$, which is definite or degenerate, not Lorentzian.