Let $X$ be a finite dimensional real Euclidean space and $S,T$ be symmetries with respect to $n-1$-dimensional subspaces of $X$.
Is it possible to write $ST$ as a composition of symmetries with respect to $(n-2)$-dimensional subspaces of $X$ ?How to do it?
Edit
A mapping $S:X\rightarrow X$ we call a symmetry if $S$ is linear and $S^2=Id$. If the set of all fixed points of $S$ is $Y$ we say that $S$ is a symmetry with respect to (a linear subspace) $Y$.
For arbitrary symmetry $S$ we have that $X$ is a prime sum of two orthogonal subspaces $X=Y+Z,$ where $Y=\{s\in X: Sx=x\}$, $Z=\{x\in X: Sx=-x\}$.
Then
$$
S(y+z)=y-z \textrm{ for } y\in Y, z\in Z.
$$