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](http://arxiv.org/pdf/1203.1901v4.pdf).