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.
Thanks in advance