Special orthogonal, special unitary, and symplectic groups as products of Grassmannians

We describe a curious structure of the special orthogonal, special unitary, and symplectic groups that has not been observed, namely, they can be expressed as matrix products of their corresponding Grassmannians realized as involution matrices. We will show that $\mathrm{SO}(n)$ is a product of two real Grassmannians, $\mathrm{SU}(n)$ a product of four complex Grassmannians, and $\mathrm{Sp}(2n,\mathbb{R})$ or $\mathrm{Sp}(2n,\mathbb{C})$ a product of four symplectic Grassmannians over $\mathbb{R}$ or $\mathbb{C}$ respectively.