Where can I find information on root systems where the inner product is other than the standard (all positive) signature?

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Lie Groups and Lie Algebrasfor more on this. It is proved there that $V=k\otimes_{\mathbf{Q}}V_0$ for the $\mathbf{Q}$-span $V_0$ of $\Phi$, irreducible decomposition is unique in a strong sense, and $V$ isabsolutely irreducibileas a $W(\Phi)$-representation for irreducible $(V,\Phi)$. Thus, if $(V,\Phi)$ is irreducible then $V$ has a $W(\Phi)$-invariant non-degenerate quadratic formuniqueup to scaling! $\endgroup$6more comments