Is there any correspondence between Lie rings and Lie groups such that if one proves for a Lie algebra $g$ with ideal $I$ that $g/I$ is a Lie algebra, then the same result holds automatically for the quotient ring?


Thanks in advance AB



Quotient of Lie rings and quotient of Lie groups!
The question is not posed in a clear way but, if I am interpreting correctly, it is enough to recall that the universal enveloping algebra $U(g/I)$ of the Lie algebra quotient $g/I$ is isomorphic to $U(g)/B$, where $B$ is the two-sided ideal of $U(g)$ generated by $I$. (Of course, here $g$ is identified with its isomorphic image in $U(g)$.) This is a rather elementary fact: see e.g. Theorem 1 in Chapter 5 of the book "N. Jacobson: Lie algebras". 


