The Killing form and maximal toral subalgebra of the complete Lie algebra (Q1352513)

Let \({\mathcal G}\) be a finite-dimensional complete Lie algebra over an algebraically closed field of characteristic zero, \({\mathcal T}\) a maximal toral subalgebra of \({\mathcal G}\) and \(\Delta\) the set of nonzero roots of \({\mathcal T}\) in \({\mathcal G}\). Let \(\langle\;,\;\rangle\) denote the restriction of \({\mathcal G}\) to \({\mathcal T}\). In this paper, the authors prove the following results: (i) \(\langle\;,\;\rangle\) is nondegenerate; (ii) Identify \({\mathcal T}^*\) with \({\mathcal T}\) via \(\langle\;,\;\rangle\). Then \(\langle\;,\;\rangle\) is positive definite on \(Q\Delta\) with values in \(Q\); (iii) \({\mathcal G}\) is simple complete if and only if \(\Delta\) is not a union of proper subsets \(\Delta_1\) and \(\Delta_2\) such that \(\langle\Delta_1, \Delta_2\rangle=0\).