Let $\pi$ be a regular, algebraic, essentially self-dual cuspidal automorphic representation of ${\mathrm{GL}}_n\left({𝔸}_F\right)$, where $F$ is a totally real field and $n$ is at most $5$. We show that for all primes $l$, the $l$-adic Galois representations associated to $\pi$ are irreducible, and for all but finitely many primes $l$, the mod $l$ Galois representations associated to $\pi$ are also irreducible. We also show that the Lie algebras of the Zariski closures of the $l$-adic representations are independent of $l$.