# Algebraic condition that distinguishes embedded from immersed lie subgroups

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?

• 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.) – José Figueroa-O'Farrill Dec 7 '19 at 23:28
• Thank you all for the answers and useful comments. – Amr Dec 8 '19 at 0:52