Let $G$ be a matrix Lie group and let $H$ be a Lie subgroup of $G$. My understanding is that all $g \in G$ are matrices, since $G$ is a subgroup of the general linear group. Then aren’t all elements $h \in H$ matrices, since $h \in G$?
From what I can gather $H$ is a matrix Lie group if $H$ is a closed subset of $G$. But if $H$ is not a closed subset of $G$, why is it not a matrix Lie group? Is the reasoning in the first paragraph wrong?
A matrix Lie group $G$ is a subgroup of the general linear group such that the limit of any convergent sequence $\{a_i\} \in G$ is in $G$. This is different than a Lie group consisting of matrices.
For example, the Lie group of all matrices with rational elements is not a matrix Lie group, since there exists a sequence of matrices that converges to a matrix with non-rational elements.