Universal Cartan extension (Q1278055)

The notion of ``complete Cartan extension'' considered in the article under review, in the reviewer's understanding, is the same as that of ``universal graded Lie algebra'' [in \textit{Q. Y. Fei} and \textit{G. Y. Shen}, J. Algebra 152, 439-453 (1992; Zbl 0781.17009)] (the formulation of the definition in the beginning of Section 3 is somewhat ambiguous and seems to contain some errors or misprints). The author shows that the ``universal graduated Lie algebra'' (which is different from Fei and Shen's ``universal graded Lie algebra'' in meaning) [\textit{I. L. Kantor}, Tr. Sem. Vektor. Tenzor. Anal. 15, 227-266 (1970; Zbl 0221.17008)] is the complete Cartan extension of the free Lie algebra generated by the linear space \(X\). In Sib. Mat. Zh. 35, 793-800 (1994; Zbl 0857.17002), the author gave a construction of the Lie algebra from a bush (of the graph theory). In the article under review, he shows that any transitive graded Lie algebra can be expressed as a quotient of a bush Lie algebra. He also gives, without proof, a classification of complete Cartan extensions \(L(r,q)\) such that the negative part of \(L(r,q)\) is \(L/L^{q+1}\) where \(L\) is a free Lie algebra generated by a space of dimension \(r\) and \(L^{q+1}\) is the sum of grading spaces of \(L\) with grades greater than \(q\). Reviewer's remark: \(L(2,4)\) is equivalent to the \(U(\Gamma^-)\) [\textit{Shen Guangyu}, Nova J. Algebra Geom. 2, 217-243 (1993; Zbl 0878.17018)]. However, the results are incompatible for characteristics \(p=2\) and 3 (\(\dim L(r,q)= \infty\) whereas \(\dim U(\Gamma^-)= 14\)).