On automorphisms of free center-by-metabelian and nilpotent groups and Lie algebras

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Let [Formula: see text] be a field of characteristic zero. For positive integers [Formula: see text] and [Formula: see text], with [Formula: see text], let [Formula: see text] be a free center-by-metabelian and nilpotent Lie algebra over [Formula: see text] of rank [Formula: see text] and class [Formula: see text], freely generated by a set [Formula: see text]. It is shown that the automorphism group [Formula: see text] of [Formula: see text] is generated by the general linear group [Formula: see text] and two more IA-automorphisms. Let [Formula: see text] be the field of rational numbers. We give [Formula: see text] the structure of a group, say [Formula: see text], via the Baker–Campbell–Hausdorff formula. Let [Formula: see text] be the subgroup of [Formula: see text] generated by [Formula: see text]. We prove that the subgroup of [Formula: see text] generated by the tame automorphisms [Formula: see text] and three more IA-automorphisms of [Formula: see text] has finite index in [Formula: see text]. For [Formula: see text], the subgroup of [Formula: see text] generated by the tame automorphisms [Formula: see text] and two more IA-automorphisms of [Formula: see text] has finite index in [Formula: see text]. A similar result is proved for the automorphism group of a free center-by-metabelian and nilpotent group of rank [Formula: see text] and class [Formula: see text].