Let $L$ be a finite dimensional Lie algebra over $\mathbb{R}$, and $K$ a subalgebra of $L$. Then, by Lie's correspondence theorems, there exists a unique (up to isomorphisms) simply connected Lie group $G$ having $L$ as a Lie algebra. There also exists a unique connected Lie subgroup $H$ of $G$ having $K$ as its Lie algebra.

Question: What algebraic condition on the pair $(L, K)$ determines if $H$ will be an embedded submanifold or not?

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    $\begingroup$ As you can see from Igor's answer, you are asking whether the connected subgroup generated by K is closed, but I'm afraid that there is no known (to me, but I've asked) purely algebraic condition on (L,K) which will guarantee this; although there are some known sufficient conditions (also listed in Igor's answer.) $\endgroup$ – José Figueroa-O'Farrill Dec 7 '19 at 23:28
  • $\begingroup$ Thank you all for the answers and useful comments. $\endgroup$ – Amr Dec 8 '19 at 0:52

See the closed-subgroup theorem, in particular the conditions for being closed.

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    $\begingroup$ Also, a further list of 8 sufficient conditions for being closed in Helgason (1978). $\endgroup$ – Francois Ziegler Dec 7 '19 at 23:25

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