Involutive Lie algebras graded by finite root systems and compact forms of IM algebras (Q1358310)

An intersection-matrix (IM) Lie algebra is defined through an analogue of the Chevalley basis of a simple Lie algebra associated with an intersection matrix \(A= (a_{ij})_{n\times n}\) \((a_{ii}=2\), \(a_{ij}<0 \iff a_{ji}<0\), \(0<a_{ij} \iff 0<a_{ji}\)). For any IM algebra, there exists an analogue of the Chevalley involution. The author shows that the fixed-point subalgebra of this involution is isomorphic to a certain Steinberg unitary Lie algebra which was introduced by \textit{B. N. Allison} and \textit{J. R. Faulkner} [J. Algebra 161, 1-19 (1993; Zbl 0812.17002)]. Moreover, the author introduces an analogue of the elementary unitary Lie algebra of other types and studies involutive Lie algebras graded by finite root systems.