The Baer–Suzuki theorem for groups of 3-exponent 1

We prove a theorem that generalizes Lemma 4 in the paper of A. S. Mamontov [Sib. Math. J., 54, No. 1, 114—118 (2013)] concerning the validity of the Baer–Suzuki theorem in groups of period 12. The results of the present paper can be used in studying groups with a given set of element orders, also called a spectrum.