Let $V$ be an $n$-dimensional real vector space and \begin{equation} \eta\colon V\times V\to\mathbb R \end{equation} a nondegenerate symmetric bilinear form. Sylvester’s Law of Inertia allows us to consider the natural number $r\in\{0,\ldots,n\}$ defined as the maximum dimension among all subspaces on which the restriction of $\eta$ is positive definite.
Question: Can there be two different $r$-dimensional subspaces of $V$ such that the restriction of $\eta$ to them is positive-definite? (Here is a related question.)
Motivation: I am currently studying special relativity and I was wondering if the tangent (translation) space has a unique decomposition as the direct sum of $1$-dimensional and a $3$-dimensional subspace.
Yes. Consider $n=2$, and $\eta(x,y) = x_1y_1-x_2y_2$. Then clearly $r=1$, and $\eta$ is positive definite over e.g. $\text{span}\{(1,0)\}$ and $\text{span}\{(2,1)\}$.
From a different point of view, there might be more than one way to diagonalize a matrix $\eta$ through $\eta\mapsto A^t \eta A$.
In terms of General Relativity, this is exactly the point: You cannot distinguish a space coordinate from a time one. Even in Special Relativity, when you do a Lorentz transformation, space and time coordinates are mixed.