I am trying to understand universal enveloping algebra and its necessity.
If $A$ is an associative algebra over a field $F$ with identity $1$, then we can form a Lie algebra $A_L$ to be $(A;+;[,])$ where $A$ is the underlying set of algebra $A$, $+$ denotes the addition operation in $A$, and $[x,y]:=xy-yx$ for $x,y\in A$. One may come up with following question: Does every Lie algebra arises in this way from an associative algebra? I am trying to understand whether universal enveloping algebra has been introduced with this question.
Q.1 Is universal enveloping algebra introduced to solve the above problem (i.e. given Lie algebra, is there associative algebra with $1$ whose associated Lie algebra is isomorphic to $L$?)
Q.2 What are the basic objects in Lie algebra to understand which we need to know universal enveloping algebra?
(My question 2 is little vague; I am simply thinking following - concerning the book of Humphrey's, for what topics the knowledge of universal enveloping algebra is necessary?)