In this paper, the author studies a question which is analogous to the problem of the existence of minimal cones arising in connection with the so-called spherical Bernstein problem [cf. W.-Y. Hsiang, Ann. Math., II. Ser. 118, 61-73 (1983; Zbl 0522.53051)]. Let \(G\) be a simple classical Lie group, and let \(S\) be the unit sphere contained in the Lie algebra of \(G\), endowed with the canonical Euclidean norm. Then there is a unique principal orbit \(A\) of the adjoint action of \(G\) which is a minimal submanifold of \(S\). \(A\) is of maximal volume in the class of all principal orbits. In the case of SU\((n)\), \(\text{SO}(n)\), \(\text{Sp}(n)\) the cone \(CA\) is a minimal submanifold. The stability of the cone \(CA\) depends on the number \(n\). See earlier results of the author for a version of this paper without proofs [Russ. Math. Surv. 41, No. 6, 201- 202 (1986); translation from Usp. Mat. Nauk 41, No. 6, 165-166 (1986; Zbl 0618.53048)].