We have the following criterion for the irreducibility of a Lie algebra representation (we work with $L$-modules here). Let $L$ be a Lie algebra, $V$ a finite dimensional vector space, and let $L \times V \to V$ be an $L$-module. Then $V$ is irreducible $\iff$ every nonzero $v \in V$ generates all of $V$.
The $\Leftarrow$ part is done. The thing I'm having trouble with is seeing that if $v \in V$ is nonzero, then there must be some $x \in L$ such that $x.v \neq 0$.