$\DeclareMathOperator\SL{SL}$Let $ G $ be a noncompact simple Lie group. For example $ \SL_n $. Let $ \Gamma $ be a lattice in $ G $. Consider the action of $ \Gamma $ on the Lie algebra of $ G $ by conjugation. Is this representation of $ \Gamma $ always irreducible?

For example, I think it is true that all lattices in $ \SL_2(\mathbb{R}) $ and $ \SL_2(\mathbb{C}) $ are Ad-irreducible.