Invariant subspace of a representation of a Lie algebra

Question

This may be silly. Let $\mathfrak{g}$ be an Lie algebra, with $V$ a nonzero representation. Denote the invariant subspace $V^{\mathfrak{g}}=\{v\in V|x.v=0,\forall x\in \mathfrak{g}\}$.

Question:Is $V^{\mathfrak{g}}$ a subrepresentation of $V$? It seems to satisfy all axioms to become a subrepresentation. But I just wonder why the textbook is vague about it (namely, Lorentz's book) and calling it a subspace.