$({\mathrm{\Theta }}_n,s{l}_n)$-graded Lie algebras $(n=3,4)$

Let $\mathbb{F}$ be a field of characteristic zero and let $\mathfrak{g}$ be a non-zero finite-dimensional split semisimple Lie algebra with root system $\mathrm{\Delta }$. Let $\mathrm{\Gamma }$ be a finite set of integral weights of $\mathfrak{g}$ containing $\mathrm{\Delta }$ and $\{0\}$. Following [2,10], we say that a Lie algebra $L$ over $\mathbb{F}$ is \emph{generalized root graded}, or more exactly $(\mathrm{\Gamma },\mathfrak{g})$-\emph{graded}, if $L$ contains a semisimple subalgebra isomorphic to $\mathfrak{g}$, the $\mathfrak{g}$-module $L$ is the direct sum of its weight subspaces $L_{\alpha }$ ($\alpha \in \mathrm{\Gamma }$) and $L$ is generated by all $L_{\alpha }$ with $\alpha \mathrm{\ne }0$ as a Lie algebra. Let $\mathfrak{g}\cong sl_{n}$ and $${\mathrm{\Theta }}_n=\{0,\pm {\epsilon }_i\pm {\epsilon }_j,\pm {\epsilon }_i,\pm 2{\epsilon }_i\mid 1\le i\mathrm{\ne }j\le n\}$$ where $\{{\epsilon }_1,\dots ,{\epsilon }_n\}$ is the set of weights of the natural $s{l}_n$-module. In [9], we classify $({\mathrm{\Theta }}_n,s{l}_n)$-graded Lie algebras for $n>4$. In this paper we describe the multiplicative structures and the coordinate algebras of $({\mathrm{\Theta }}_n,s{l}_n)$-graded Lie algebras $(n=3,4)$. In $n=3$, we assume that $$[V(2{\omega }_1)\otimes C,V(2{\omega }_1)\otimes C]=[V(2{\omega }_2)\otimes C^{\mathrm{\prime }},V(2{\omega }_2)\otimes C^{\mathrm{\prime }}]=0$$ where $V(\omega )$ is the simple $\mathfrak{g}$-module of highest weight $\omega$, $C={\rm Hom_{\mathfrak{g}}}(V(2\omega_{1}),L)$and$C’={\rm Hom_{\mathfrak{g}}}(V(2\omega_{2}),L)$ .