Algebr Represent Theor (2010) 13:653–660 DOI 10.1007/s10468-009-9165-2 Restricted Enveloping Algebras with Minimal Lie Derived Length Francesco Catino · Salvatore Siciliano · Ernesto Spinelli Received: 11 September 2008 / Accepted: 1 July 2009 / Published online: 2 December 2009 © Springer Science + Business Media B.V. 2009 Abstract Let L be a non-abelian restricted Lie algebra over a field of characteristic p > 0 and let u( L) denote its restricted enveloping algebra. In Siciliano (Publ Math (Debr) 68:503–513, 2006) it was proved that if u( L) is Lie solvable then the Lie derived length of u( L) is at least ⌈log 2 ( p + 1)⌉. In the present paper we characterize the restricted enveloping algebras whose Lie derived length coincides with this lower bound. Keywords Restricted Lie algebra · Enveloping algebra · Lie derived length AMS 2000 Subject Classifications 16S30 · 17B50 · 17B60 1 Introduction Let R be a unital associative algebra over a field F . Then R can be regarded as a Lie algebra via the Lie multiplication defined by [x, y] := xy − yx, for all x, y ∈ R. For subspaces A, B ⊆ R, we denote by [ A, B] the linear span of all elements [a, b ], with a ∈ A and b ∈ B. The Lie derived series of R is defined inductively by δ [0] ( R) := R Presented by Donald Passman. F. Catino (B ) · S. Siciliano · E. Spinelli Dipartimento di Matematica “E. De Giorgi”, Università del Salento, Via Provinciale Lecce-Arnesano, 73100 Lecce, Italy e-mail: francesco.catino@unisalento.it S. Siciliano e-mail: salvatore.siciliano@unisalento.it E. Spinelli e-mail: ernesto.spinelli@unisalento.it