# query about Jacques Tits' “Uniqueness and Presentation of Kac-Moody groups over fields”

In his paper on Kac-Moody groups Jacques Tits uses $\Phi$ to represent the set of real roots of a Kac-Moody algebra and in Section 3.7(a) he writes "Let $\Phi_{1}$ be a finite semisimple closed subsystem of $\Phi$." What does he mean by "semisimple"?

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Taken in isolation the question is reasonable, but it's important to add the explicit journal reference: J. Algebra 105 (1987), 542-573. This paper is rather technical and necessarily heavy on notation, starting with the usual definition of generalized Cartan matrix and corresponding Kac-Moody Lie algebra over $\mathbb{C}$ given by generators and relations. Tits usually refers just to "roots" or "real roots", the images of the fixed "simple" roots under the "Weyl group", since imaginary roots play little role in the paper. While he gives the usual definition of "closed" set of roots in 3.2 (a set containing the sums of any two of its roots which are again roots), he does not define explicitly what it means to be a finite "semisimple" closed subsystem. From the context, this would be a finite closed set of roots which is the usual root system of a finite dimensional semisimple subalgebra of the given Kac-Moody algebra (as Koen S suggests).