  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
$\varepsilon:\Gamma\to\mathrm{PG}(V)$ 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 $\varepsilon$-preimage of a projective subspace of $\mathrm{PG}(V)$.
