Leibniz algebras and graphs

We consider a Leibniz algebra $\mathfrak{L}=\mathfrak{I}\oplus \mathfrak{V}$ over an arbitrary base field $\mathbb{F}$, being $\mathfrak{I}$ the ideal generated by the products $[x,x],x\in \mathfrak{L}$. This ideal has a fundamental role in the study presented in our paper. A basis $\text{B}=\{v_i{\}}_{i\in I}$ of $\mathfrak{L}$ is called multiplicative if for any $i,j\in I$ we have that $[v_i,v_j]\in \mathbb{F}{v}_k$ for some $k\in I$. We associate an adequate graph $\mathrm{\Gamma }(\mathfrak{L},\text{B})$ to $\mathfrak{L}$ relative to $\text{B}$. By arguing on this graph we show that $\mathfrak{L}$ decomposes as a direct sum of ideals, each one being associated to one connected component of $\mathrm{\Gamma }(\mathfrak{L},\text{B})$. Also the minimality of $\mathfrak{L}$ and the division property of $\mathfrak{L}$ are characterized in terms of the weak symmetry of the defined subgraphs $\mathrm{\Gamma }(\mathfrak{L},{\text{B}}_{\mathfrak{I}})$ and $\mathrm{\Gamma }(\mathfrak{L},{\text{B}}_{\mathfrak{V}})$.