G is a connected compact Lie group and G' is a dense Lie subgroup of G, they correspond to Lie Algebras $\mathfrak{g}$ and $\mathfrak{g'}$. Now we have questions:
(I)Can we proof $\mathfrak{g'}$ is an ideal of $\mathfrak{g}$?
(II)How to proof $\mathfrak{g'}$ is communicative if and only if $\mathfrak{g}$ is communicative?
(III)Can we proof there is a communicative ideal $\mathfrak{g''}$ of $\mathfrak{g}$ such that we have a direct sum decomposition of Lie Algebra: $\mathfrak{g}=\mathfrak{g'}+\mathfrak{g''} $?
Can we proof G'=G? I do not know how to proof it so I want too see a full proof.