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Let $k$ be a field of characteristic $0$. We introduce a pair of adjointfunctors, Allison-Benkart-Gao functor $\mathcal{ABG}$ and Berman-Moody functor$\mathcal{BM}$, between the category of non-unital alternative algebras over$k$ and the category ${\text{\bf Lie}_{\text{R}}}$ of Lie algebras withappropriate $sl_3(k)$-module structures. Surprisingly, when $A$ is a non-unitalalternative algebra, the Allison-Gao Lie algebra $\mathcal{ABG}(A)$ isdifferent from the more well-known Steinberg Lie algebra $st_3(A)$.  Next, let $A(D)$ be the free (non-unit) alternative algebra generated by $D$elements and $\text{Inner} A(D)$ the inner derivation algebra of $A(D)$. Aconjecture on the homology of $H_r(\mathcal{ABG}(A(D)))$ is proposed.  Let $A(D)_n$(resp. $\text{Inner} A(D)_n$) be the degree $n$ component of$A(D)_n$(resp. $\text{Inner} A(D)_n$). The previous conjecture implies anotherconjecture on the dimensions on $A(D)_n$ and $\text{Inner} A(D)_n$. We alsogive some evidences to support the these conjectures.

Allison-Benkart-Gao functor and the free non-unital alternative algebras