Nearly all subspaces of a classical polar space arise from its universal embedding

Let $Gamma$ be an embeddable non-degenerate polar space of finite rank $n geq 2$. Assuming that $Gamma$ admits the universal embedding (which is true for all embeddable polar spaces except grids of order at least $5$ and certain generalized quadrangles defined over quaternion division rings), let $arepsilon:Gamma omathrm{PG}(V)$ be the universal embedding of $Gamma$. Let $cal S$ be a subspace of $Gamma$ and suppose that $cal S$, regarded as a polar space, has non-degenerate rank at least $2$. We shall prove that $cal S$ is the $arepsilon$-preimage of a projective subspace of $mathrm{PG}(V)$.