Let $G$ be a complex linear algebraic group with Lie algebra $\mathfrak{g}$. A subalgebra $\mathfrak{h}$ in $\mathfrak{g}$ is called algebraic if the unique connected Lie subgroup $H$ of $G$ with Lie algebra $\mathfrak{h}$ is an algebraic subgroup of $G$ (i.e. Zariski-closed).
Question: Does this definition depend on the group $G$?
In other words, if $\mathfrak{h}$ is algebraic, and $\tilde{G}$ is another linear algebraic group with Lie algebra $\mathfrak{g}$, is $\mathfrak{h}$ algebraic with respect to $\tilde{G}$?