Let $\mathfrak{g}$ be a semisimple Lie algebra, $V$ be a vector space, and $R:\mathfrak{g}\rightarrow \mathfrak{gl}(V)$ be a representation of $\mathfrak{g}$. Let $\{X\}\subsetneqq\mathfrak{g}$ be a generating(in the Lie algebra sense) subset of $\mathfrak{g}$, i.e. $\mathfrak{g} = \langle \{X\} \rangle$. Let $\mathfrak{h}$ be a strict Lie subalgebra generate by a subset of $\{X\}$, and denote by $H$ all such $\mathfrak{h}$. Then is the following statement true?
If the representation $R$ restricted on $\mathfrak{h}$ is faithful for all $\mathfrak{h}\in H$, then $R$ is a faithful representation of $\mathfrak{g}$.