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Let φ be a Jordan automorphism of an algebra A. The situation when an element a ∈ A satisfies 2 (φ(a) + φ−1(a)) = a is considered. The result which we obtain implies the Kleinecke–Shirokov theorem and Jacobson’s lemma. Let A be a finite-dimensional algebra over a field of characteristic 0. Jacobson’s lemma [4, Lemma 2] from 1935 asserts, in its original form, that if a, b ∈ A are such that

A Kleinecke–Shirokov type condition with Jordan automorphisms