Let $V=(V,b)$ be a finite dimensional vector space equipped with $b$ a symmetric and positive definite bilinear form. And let $\{e_1,\dotsc,e_n\}$ be a orthonormal basis for the subspace $\ker(p_A)$ ($p_A$ is defined below).

For a matrix $A \in \mathrm{O}(V)$, let $\mathrm{O}_*(V)$ the subset of $\mathrm{O}(V)$ such that be the matrix $P_A:=\frac{A-JAJ}{2}$ is invertible, where $J$ is a complex structure (a matrix such that $J^2=-1$).

Let $n=\dim \ker(P_A)$. For every $j \in \{1,\dotsc,n\}$ we define the reflexions $r_j$ such that $r_j(e_j)=-Je_j$, $r(Je_j)=-e_j$ and $r_j(v)=v$ for any $v \in V$ such that $b(v,e_j)=b(v,Je_j)=0$. Finally, let $$R:=r_1r_2\dotsm r_n \in \mathrm{O}(V).$$

I need to prove that $$RA \in \mathrm{SO}_*(V),$$ where similarly as $\mathrm{O}(V)$: $\mathrm{SO}_*(V)$ is the subset of $\mathrm{SO}(V)$ such that $P_B:=\frac{B-JBJ}{2}$ is invertible.

I already proved that $RA \in \mathrm{SO}(V)$; the only thing that I haven’t been able to figure out is to prove that $\frac{1}{2}(RA-JRAJ)$ is invertible, since $n$ can be even or odd.

Also, $P_{r_j}$ is not invertible since $\det(r_j)=-1$.

What is good and optimized approach to deal with the product of reflections $$R=r_1r_2\dotsm r_n?$$

Any help will be greatly appreciated.