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The Frobenius-Schur indicator (of a self-dual finite dimensional representation) is $$ \chi_\pi(\exp(2\pi i\rho^\vee)) $$ where $\chi_\pi$ is the central character of $\pi$, $\rho^\vee$ is half the sum of the positive coroots, so $\exp(2\pi i\rho^\vee)$ is an element of order $2$ in the center of $G$. Equivalently if $\lambda$ is the highest weight then this equals $$ e^{2\pi i\langle\lambda,\rho^\vee\rangle} $$ This is, of course, equivalent to the other formulas cited, but is conceptually simpler. In particular: if $G$ is adjoint every (irreducible, finite dimensional, self-dual) representation is orthogonal. See Bourbaki, Lie Groups and Lie Algebras, Chapters 7-9, Chapter IX, Section 7.2, Proposition 1. The proof is included, and is the one sketched by Borovoi. For a simpler proof based on the Tits group see this preprint.