Let $\Sigma$ be a three dimensional submanifold of the lorentzian manifold $(M,g)$ and let $q$ the riemannian metric on $\Sigma$ induced by $g$. We define the Weingarten mapping as $$ W:T\Sigma \rightarrow T\Sigma \qquad X\mapsto W(X)={}^4\nabla_X n$$ such that $n$ is normal to $\Sigma$ within $(M,g)$ and ${}^4\nabla$ is the levi-civita connection for $g$ on $M$.
Question why is $W$ well defined i.e: $W(X) \in T\Sigma$ ? As I saw we have $$ \langle W(X), n\rangle = \langle {}^4\nabla_X n,n\rangle =\frac{1}{2}X\langle n,n \rangle =0$$
The second equality follows from ${}^4\nabla$ being compatible with $g$. But why $X\langle n,n \rangle =0$ ?