Ideals and primitive elements of some relatively free Lie algebras

Let F be a free Lie algebra of finite rank over a field K. We prove that if an ideal 〈v~ 〉 of the algebra F/ ?m+1(F') contains a primitive element u~ then the element v~ is primitive. We also show that, in the Lie algebra F/ ?3(F) ' there exists an element v¯ such that the ideal 〈v¯ 〉 contains a primitive element u¯ but, u¯ and v¯ are not conjugate by means of an inner automorphism. © 2016, The Author(s)