In the map $\phi : L \mapsto \mathfrak {U}(L) $, where $ L $ is a lie algebra and $\mathfrak {U} $ is a universal enveloping algebra of $ L $.
(1) Is the following relation true?
If $[xy]=z$ in $ L $, does $\phi (x) \phi (y)-\phi (y) \phi (x) = \phi (z) $?
(2) In addition if all commutators are zero in the symmetric tensor algebra of $ L $, are all brackets zero too?
Help is really appreciated.