Let A be the free non-associative algebra and let T be the T -ideal generated by the identity (x, y, z)+( y, x, z). Given an ideal J ⊇ T , then CJ = A/J is a left alternative algebra. We construct an ideal ΓJ and we define a universal enveloping algebra of CJ as A/ΓJ . We introduce a linear map ω : CJ → A/ΓJ , such that ω((a, b, c)) = (a, b, c)−(b, a, c). As a conjecture we state that ω is injective. The injection of CJ into A/ΓJ is similar to the injection of a Lie algebra into an associative algebra by [a, b ]= ab − ba; moreover we show how to construct a spanning set of A/ΓJ from a basis of CJ and we define a universal property analogously like in the case of Lie algebras.