Proportionality of the Lie words in the classical Lie algebras (Q1397548)

Let \(R\) be a reduced root system of rank \(r\) in Euclidean space \(E\). Let \(\Pi = \{\alpha_1,\dots,\alpha_r\}\) be some subsystem of simple roots and \(R^+\cup R^-\) be the partition of \(R\) on positive and negative parts respect to \(\Pi\). A Lie algebra \(L=L^+\) over an arbitrary field \(K\) is called an algebra of type \(R^+\) if \(L\) has the decomposition \(\oplus_{\alpha \in R^+}L_\alpha\) satisfying the following conditions: 1) \((\forall \alpha\in R^+) \dim L_\alpha \geq 1\); 2) \((\forall \alpha, \beta \in R^+)\) \([L_\alpha, L_\beta] \subseteq L_{\alpha +\beta}\) (if \(\alpha +\beta \notin R^+\) then \([L_\alpha, L_\beta]=0\)); 3) \(\dim L_{\alpha_i} = 1\) \((i=1,2,\dots,r)\) and \(L\) is generated by the subspace \(\oplus_{i=1}^rL_{\alpha_i}\). By analogue, the algebras of types \(R^-\) and \(R\) are defined. Let \(L\) be a Lie algebra of type \(R^+\). For each \(\alpha_i\in \Pi\), some nonzero vector \(f_i\) of the root space \(L_i=Kf_i\) \((i=1,2,\dots,r)\) is fixed. Let \(a=(i_1, i_2,\dots,i_m)\), \(1\leq i_s \leq r\) be an ordered sequence of indices, \(\varphi_a\) be the weight \(\varphi_a= \alpha_{i_1}+\alpha_{i_2}+\cdots +\alpha_{i_m}\), and \(f_a= [f_{i_1},f_{i_2}, f_{i_3},\dots,f_{i_m}]\). The sequence \(a=(i_1, i_2,\dots,i_m)\) is called correct if \(\alpha_{i_1}+\alpha_{i_2}+\cdots +\alpha_{i_s} \in R\) for any \(s=1,2,\dots,m\). The oriented graph with coloured arcs (by indices \(1\leq i_s \leq r\)) is assigned to the set \(R^0=R\cup\{0\}\) of the following way: vertices of the graph are elements of \(R^0\), two vertices are joined by the arc with the begining \(\alpha\), the end \(\beta\) and index \(i\) iff \(\beta=\alpha + \alpha_i\). This graph is called chema of \(R\) with respect to \(\Pi\). Using some number characteristics of this chema and Dynkin's chema, the author proves that if \(a=(i_1, i_2,\dots,i_m)\) and \(b=(j_1, j_2,\dots,j_m)\) are correct sequences and \(\varphi_a= \varphi_b\) then in \(L^+\) words \(f_a\) and \(f_b\) are proportional (Theorem 1). Moreover the author finds necessary and sufficient conditions such that algebra \(L=L^-\oplus H \oplus L^+\) is an algebra of type \(R\) (in the same reduced Kartan's matrices is used) (Theorem 2).