$g$-equivalent subspaces

The following definition is from the book Lectures in Abstract Algebra, by Jacobson. Chapter "Witt's theorem".

Definition Let $$\Re$$ be a vector space, let $$\mathfrak{S}_1$$ and $$\mathfrak{S}_2$$ be its subspaces. Then subspaces $$\mathfrak{S}_1$$ and $$\mathfrak{S}_2$$ are $$g$$-equivalent if there exist a bijective linear map $$U: \mathfrak{S}_1 \rightarrow \mathfrak{S}_2$$ such that $$g(x_1,y_1) = g(U(x_1),U(y_1))$$ for all $$x_1,y_1 \in \mathfrak{S}_1$$.

Question: I tried to look for an examples of $$g$$-equivalent subspaces but I could not find anything, even the definition. Could someone please give an example of a vector space, its $$g$$-equivalent subspaces and a bijective linear map so that the definition holds.

• For example, if $g$ is a positive definite bilinear form (inner product), then any two $k$ dimensional subspaces are $g$-equivalent. It's not true, however, for semidefinite or indefinite forms. – Berci Apr 8 at 15:37
• Thank you for your answer. In this case $U$ can be any bijective linear map? – IllariyaS Apr 8 at 16:31