Page 11 of "Linear Algebra and Geometry" by Irving Kaplansky
Theorem 8: Let $V$ be a finite dimensional inner product space over an ordered field $K$. Then $V$ can be written as an orthogonal direct sum $V= W_1 \oplus W_2 \oplus W_3$, where the inner product is positive definite on $W_1$, negative definited on $W_2$ and identically zero on $W_3$. In such a decomposition the dimensions of $W_1$, $W_2$ and $W_3$ are uniquely determined.
Is there any obvious reason why the $K$ must be ordered for this theorem to hold? I mean, do the ideas of positive definite and negative definite make sense in an unordered field? I think so, you can still know which elements are negative and which are positive even if the field is not ordered, right? The fact that $K$ is ordered never appears in the proof of Theorem 8.