Let $G$ be a compact, simply connected Lie group, and $H$ is its Lie subgroup that is also compact and simply connected, and has the same dimension with $G$, then should $H=G$? Note that these imply that the Lie group $G$ and $H$ are locally isomorphic (consider a small neighborhood around the identity).
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The answer is yes. Quoting from Sagle and Walde's 'Introduction to Lie Groups and Lie Algebras':
Corollary 8.7: Let $G$ and $H$ be simply connected Lie groups with Lie algebras $g$ and $h$. If $g$ and $h$ are isomorphic Lie algebras, then $G$ and $H$ are isomorphic Lie groups.
This follows from
Theorem 8.6: For simply connected groups, if $f : g \rightarrow h$ is a homomorphism of the Lie algebras then there is a unique Lie group homomorphism $\psi: G \rightarrow H$ with $T\psi(e) = f$; where $T\psi(e)$ is the tangent map of $\psi$ at the identity.
In your example $G$ and $H$ have the same Lie algebra and so $G$ is isomorphic to $H$.