According to this page on Engel’s theorem on [Wikipedia] (https://en.m.wikipedia.org/wiki/Engel%27s_theorem#Proof)
Write $\mathfrak{g} = \mathfrak{h} + L $ where $L$ is a one-dimensional vector subspace. Let $Y$ be a nonzero vector in $L$ and $ v $ a nonzero vector in $W$. Now, Y is a nilpotent endomorphism (by hypothesis) and so $Y^k(v) \neq 0, Y^{k+1}(v) = 0 $for some $k$. Then $Y^k(v)$ is a required vector as the vector lies in $W$ by Step 2. $\square $
But I cannot see anywhere on that page where we can say that $W$ contains a nonzero vector.
Any help here?
The proof is by induction on the dimension of $\mathfrak{g}$. So the induction hypothesis tells you that the desired result already holds for $\mathfrak{h}$, which exactly says that $W$ contains a nonzero vector.