Serre relations for $A_2$ recover ${\mathfrak{s}\mathfrak{l}}_3$

The images satisfy the Cartan and generator relations: $[h_i,h_j]=0$; $[h_1,E_{12}]=2E_{12}$, $[h_1,E_{23}]=-E_{23}$, $[h_2,E_{12}]=-E_{12}$, $[h_2,E_{23}]=2E_{23}$ and the negatives on the $f$'s; $[E_{12},E_{21}]=h_1$, $[E_{23},E_{32}]=h_2$, and the cross brackets $[E_{12},E_{32}]$ and $[E_{23},E_{21}]$ vanish.

Direct multiplication also gives $(\mathrm{ad}{E}_{12}{)}^2{E}_{23}=(\mathrm{ad}{E}_{23}{)}^2{E}_{12}=(\mathrm{ad}{E}_{21}{)}^2{E}_{32}=(\mathrm{ad}{E}_{32}{)}^2{E}_{21}=0$, so all four Serre relations hold. Hence the assignment induces a Lie-algebra homomorphism $\phi :\mathfrak{g}(A)\to {\mathfrak{s}\mathfrak{l}}_3(\mathbb{C})$.

The images generate ${\mathfrak{s}\mathfrak{l}}_3(\mathbb{C})$: $[E_{12},E_{23}]=E_{13}$, $[E_{23},E_{12}]=-E_{13}$, and $[E_{21},E_{32}]=-E_{31}$. Thus the image contains all six off-diagonal matrix units and the independent diagonal matrices $h_1,h_2$, which form a basis of the eight-dimensional space of traceless $3\times 3$ matrices. Hence $\phi$ is surjective.

Put $z=[e_1,e_2]$. The positive Serre relations give $[e_1,z]=[e_2,z]=0$, so $\mathrm{span}(e_1,e_2,z)$ is a Lie subalgebra containing the positive generators and contained in the subalgebra they generate; hence it equals ${\mathfrak{n}}_{+}$. With $w=[f_1,f_2]$, the negative Serre relations likewise give $[f_1,w]=[f_2,w]=0$, so ${\mathfrak{n}}_{-}=\mathrm{span}(f_1,f_2,w)$ and each half has dimension at most three. Since $\mathfrak{h}$ is spanned by $h_1,h_2$, the triangular decomposition in [L1] gives $\dim \mathfrak{g}(A)\le 8$. Surjectivity from step 3.1 onto the eight-dimensional algebra ${\mathfrak{s}\mathfrak{l}}_3(\mathbb{C})$ gives the reverse inequality, so $\dim \mathfrak{g}(A)=8$ and $\ker \phi =0$. Thus $\phi$ is the asserted isomorphism.