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See the book by Onishchik and Vinberg "Lie Groups and Algebraic Groups", Springer 1990, page 297, Table 3. For any irreducible root system, the last column of the table gives an element $b\in P^\vee$. Onishchik and Vinberg write that the self-dual irreducible representation with highest weight $\lambda$ admits a symmetric invariant bilinear form if and only if $\lambda(b)\in\mathbb{Z}$. This is the same as to say that a certain linear combination with integer coefficients (described in the table) of the numbers $\lambda_i$ is even. In other words, the Frobenius-Schur indicator is $-1$ to the power of this linear combination.