Irreducible representation of a Lie algebra

Let $\mathfrak g$ be a Lie algebra and let $\rho:\mathfrak g\to \mathfrak{gl}(V)$ be a representation on a finite-dimensional vector space $V$.

Definition (Irreducible).
The representation $(\rho,V)$ is irreducible if the only $\mathfrak g$-invariant subspaces of $V$ are $\{0\}$ and $V$. Equivalently, $V$ is a simple $\mathfrak g$-module.

A subspace $W\subseteq V$ is $\mathfrak g$-invariant precisely when $\rho(x)W\subseteq W$ for all $x\in\mathfrak g$; such a $W$ is a subrepresentation .

Context.
Irreducibles are the building blocks for representation theory. For semisimple $\mathfrak g$, every finite-dimensional representation is completely reducible (see Weyl's theorem and complete reducibility ), and irreducibles are classified by the highest-weight theorem .