Let $\mathfrak{g}$ be a $n$-dimensional Lie algebra and $\mathfrak{h}$ be a $k$-dimensional Lie algebra ($k < n$). The multiplication tables for these Lie algebras are known. Is there a way to show that $\mathfrak{h}$ is isomorphic to a subalgebra of $\mathfrak{g}$? Or there is no "algorithm"?

Let $\mathfrak{g}$ be a $n$-dimensional Lie algebra and $\mathfrak{h}$ be a $k$-dimensional Lie algebra ($k < n$). The multiplication tables for these Lie algebras are known. Is there a way to show that $\mathfrak{h}$ is isomorphic to a subalgebra of $\mathfrak{g}$? Or there is no "algorithm"?

To be more specific, I'd like to know the answer to this question if $\mathfrak{g}$ is the 8-dimensional real Lie algebra $\mathfrak{g}$ with multiplication table

and $\mathfrak{h}$ is some 3-dimensional real Lie algebra from the Bianchi classification, e.g. $$[E_1,E_2] = E_2, [E_2,E_3] = 0, [E_3,E_1] = -E_3.$$

Let $\mathfrak{g}$ be a $n$-dimensional Lie algebra and $\mathfrak{h}$ be a $k$-dimensional Lie algebra ($k < n$). The multiplication tables for these Lie algebras are known. Is there a way to show that $\mathfrak{h}$ is a subalgebra of $\mathfrak{g}$? Or there is no "algorithm"?

Let $\mathfrak{g}$ be a $n$-dimensional Lie algebra and $\mathfrak{h}$ be a $k$-dimensional Lie algebra ($k < n$). The multiplication tables for these Lie algebras are known. Is there a way to show that $\mathfrak{h}$ is isomorphic to a subalgebra of $\mathfrak{g}$? Or there is no "algorithm"?