Read about Frobenius-Schur anywhere. In a nutshell a complex irreducible $V$ of complex dimension $n$ can give 1 or 2 real irreducibles, whose real dimensions are $n$ or $n/2$. This can be easily determined by computing the FS-indicator or dimensions of invariants in both $S^2V$ and $\Lambda^2 V$. The latter can be done in Lie for each particular representation.

Read about Frobenius-Schur anywhere. In a nutshell a complex irreducible $V$ of complex dimension $n$ can give 1 or 2 real irreducibles, whose real dimensions are $n$ or $2n$. This can be easily determined by computing the FS-indicator or dimensions of invariants in both $S^2V$ and $\Lambda^2 V$. The latter can be done in Lie for each particular representation.