Does a basis for a Lie algebra of a Lie group $G$ depend on whether $G$ is embedded?

Question

My professor recently gave the problem to find a basis for a lie algebra of a given embedded lie subgroup. The problem stressed that the lie subgroup was embedded (which was clear from the definition of the subgroup), but I cannot see that it should matter to the choice of basis. Hence this question. Does a basis for a lie algebra depend on whether the lie group is embedded somehow? Or put differently, suppose $G$ is an embedded Lie subgroup of a Lie group $M$. What is the difference between:

1. a basis for the Lie algebra of $G$, and
2. a basis for the Lie algebra of $G$ regarded as a subspace of $M$?

Answer 1

My guess is that whoever wrote the second sentence meant basis of the Lie algebra of $G$ which is a subset of the Lie algebra of $M$.