LetLbe a Lie algebra, nilpotent of class 2, over an infinite fieldK, and suppose that the centreCofLis one dimensional; such Lie algebras are called Heisenberg algebras. Let ρ:L→homKVbe a finite dimensional representation of the Heisenberg algebraLsuch that ρ(C) contains non-singular linear transformations ofV, and denotel(ρ) the ideal of identities for the representation ρ. We prove that the ideals of identities of representations containingI(ρ) and generated by multilinear polynomials satisfy the ACC. Letsl2(L) be the Lie algebra of the traceless 2×2 matrices overK, and suppose the characteristic ofKequals 2. As a corollary we obtain that the ideals of identities of representations of Lie algebras containing that of the regular representation ofsl2(K) and generated by multilinear polynomials, are finitely based. In addition we show that one cannot simply dispense with the condition of multilinearity. Namely, we show that the ACC is violated for the ideals of representations of Lie algebras (over an infinite field of characteristic 2) that contain the identities of the regular representation ofsl2(K).