Bulletin of the Australian Mathematical Society Bull. Austral. Math. Soc. 0004-9727 ALNBAB 2005 72 1 10.wxyz/CV72P1 http://www.austms.org.au/Publ/Bulletin/V72P1/ R.M. Bryant L.G. Kovács Ralph Stöhr 14 February 2006 2006 2 14 147 156 721-5118-BrKoSt-2005 10.wxyz/C2005V72P1p147 http://www.austms.org.au/Publ/Bulletin/V72P1/721-5118-BrKoSt/ A theorem independently due to A.I. Shirshov and E. Witt asserts that every subalgebra of a free Lie algebra (over a field) is free. The main step in Shirshov's proof is a little known but rather remarkable result: if a set of homogeneous elements in a free Lie algebra has the property that no element of it is contained in the subalgebra generated by the other elements, then this subset is a free generating set for the subalgebra it generates. Witt also proved that every subalgebra of a free restricted Lie algebra is free. Later G.P. Kukin gave a proof of this theorem in which he adapted Shirshov's argument. The main step is similar, but it has come to light that its proof contains substantial gaps. Here we give a corrected proof of this main step in order to justify its applications elsewhere. 17B01, 17B50 MR2162300 02212192 Yu. A. Bakhturin (Russian) Nauka MR0886063 English translation: (VNU Science Press, Utrecht, 1987) Yu. A. Bakhturin; \textit{Identical relations in Lie algebras}, (Russian) (Nauka, Moscow, 1985). English translation: (VNU Science Press, Utrecht, 1987) R.M. Bryant and R. Stöhr Trans. Amer. Math. Soc. MR1621725 R.M. Bryant and R. St\"ohr; On the module structure of free Lie algebras, \textit{Trans. Amer. Math. Soc.} \textbf{352} (2000), pp.~901--934. R.M. Bryant, L.G. Kovács and R. Stöhr Proc. London Math. Soc. (3) MR1881395 R.M. Bryant, L.G. Kov\'acs and R. St\"ohr; Lie powers of modules for groups of prime order, \textit{Proc. London Math. Soc. (3)} \textbf{84} (2002), pp.~343--374. N. Bourbaki (Part I: Chapters 1--3) Hermann MR453824 N. Bourbaki; \textit{Lie groups and Lie algebras}, (Part I: Chapters 1--3) (Hermann, Paris, 1975). N. Jacobson Interscience MR143793 N. Jacobson; \textit{Lie algebras} (Interscience, New York, 1962). L.G. Kovács and R. Stöhr J. Algebra MR1769283 L.G. Kov\'acs and R. St\"ohr; Lie powers of the natural module for $GL(2)$, \textit{J. Algebra} \textbf{229} (2000), pp.~435--462. G.P. Kukin (Russian) Algebra i Logika MR318251 English translation: Algebra Logic11 (1972), 294--303 G.P. Kukin; Subalgebras of free Lie $p$-algebras, (Russian), \textit{Algebra i Logika} \textbf{11} (1972), pp.~535--550. English translation: Algebra Logic11 (1972), 294--303 C. Reutenauer Clarendon Press MR1231799 C. Reutenauer; \textit{Free Lie algebras} (Clarendon Press, Oxford, 1993). A.I. Shirshov (Russian) Mat. Sbornik N. S. MR59892 A.I. Shirshov; Subalgebras of free Lie algebras, (Russian), \textit{Mat. Sbornik N. S.} \textbf{33} (1953), pp.~441--452. R. Stöhr J. Austral. Math. Soc. MR1847196 R. St\"ohr; Restricted Lazard elimination and modular Lie powers, \textit{J. Austral. Math. Soc.} \textbf{71} (2001), pp.~259--277. E. Witt Math. Z. MR77525 E. Witt; Die Unterringe der freien Lieschen Ringe, \textit{Math. Z.} \textbf{64} (1956), pp.~195--216.