A sum-bracket theorem for simple Lie algebras

Let $\mathfrak{g}$ be an algebra over $K$ with a bilinear operation $[\cdot,\cdot]:\mathfrak{g}\times\mathfrak{g}\rightarrow\mathfrak{g}$ not necessarily associative. For $A\subseteq\mathfrak{g}$, let $A^{k}$ the set of elements written combining $k$ $A$ via $+$ and $[\cdot,\cdot]$. We show "sum-bracket theorem" for simple Lie algebras form $\mathfrak{g}=\mathfrak{sl}_{n},\mathfrak{so}_{n},\mathfrak{sp}_{2n},\mathfrak{e}_{6},\mathfrak{e}_{7},\mathfrak{e}_{8},\mathfrak{f}_{4},\mathfrak{g}_{2}$: if $\mathrm{char}(K)$ is too small, we have growth $|A^{k}|\geq|A|^{1+\varepsilon}$ all generating symmetric sets away from subfields $K$. Over $\mathbb{F}_{p}$ in particular, diameter bound matching best analogous bounds groups type [BDH21]. As independent intermediate result, prove also estimate $|A\cap V|\leq|A^{k}|^{\dim(V)/\dim(\mathfrak{g})}$ linear affine subspaces $V$ $\mathfrak{g}$. This valid algebras, especially small large class them including associative, Lie, Mal'cev superalgebras.