I want to show that the adjoint representation of $sp(4,\mathbb{R})$ is irreducible, but I am kind of stuck.
I can assume there's some non zero subspace of $sp(4,\mathbb{R})$ which is invariant under $ad_X$, where $X \in sp(4,\mathbb{R})$, but I don't see how to show that they are equal?
Any hints?
Here are few more details for the answer given in the comment by user8268:
Let ${\mathfrak g}$ be a (finite-dimensional) Lie algebra with the Lie group $G$. Irreducibility of the adjoint action of $G$ would follow from irreducibility of the adjoint action of the Lie algebra ${\mathfrak g}$ on itself. The latter means that there are no proper subspaces $V\subset {\mathfrak g}$ so that $[v,g]\in V$ for all $v\in V, g\in {\mathfrak g}$. Such a subspace would necessarily be an ideal in ${\mathfrak g}$. A Lie algebra ${\mathfrak g}$ is called simple if it has no proper ideals. Proof of simplicity of the Lie algebra ${\mathfrak g}=sp(4)$ can be found in any book on Lie algebras.