Abstract
It is proved that the algebra of multiplications of the free commutative alternative algebra of finite rank [Formula: see text] is strongly Lie nilpotent of class [Formula: see text]. It is found the class of nilpotency of the ideal, generated by commutators in the free [Formula: see text]-generated associative algebra with identity of Lie nilpotency of degree [Formula: see text] under the condition that [Formula: see text] or [Formula: see text].
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