Understanding some notation and concepts regarding Universal Enveloping Algebras I'm learning about Universal Enveloping Algebras, and I just want to clear up my understanding of a bit of notation and confirm my understanding of a bit of the concept.
If we have a lie algebra $\mathfrak{g}$, then the Universal Enveloping algebra of $\mathfrak{g}$ is defined as:
$$U(\mathfrak{g}) = k<x_1,x_2,...,x_n >/<x_ix_j-x_jx_i-\sum_{s=1}^n C_{ij}^s x_s>$$
I am trying to be clear about the notation of: $C_{ij}^s$
I think I am correct in understanding that the Universal enveloping algebra is constructing an associative algebra by taking non-commuting polynomials in the basis elements of the lie algebra and then quotienting off the bracket relations, but I would just like to know the specific definition of $C_{ij}^s$
Secondly, reading from wikipedia, I'm a bit confused with this explanation:

Any associative algebra $A$ over the field $K$ becomes a Lie algebra over
  $K$ with the Lie bracket:
${\displaystyle [a,b]=ab-ba} .$ That is, from an associative product,
  one can construct a Lie bracket by taking the commutator with respect
  to that associative product. Denote this Lie algebra by $A_L$.
Construction of the universal enveloping algebra attempts to reverse
  this process: to a given Lie algebra $L$ over $K$, find the "most general"
  unital associative K-algebra $A$ such that the Lie algebra $A_L$ contains
  $L$; this algebra $A$ is $U(L)$.

I'm a bit confused about the relationship between $A_L$ and $L$. From the explanation, it doesn't seem like $L$ has any relationship with $A_L$ to begin with, but then it says $A_L$ contains $L$?
What's going on here? Thanks for any help!
 A: You use a basis-dependent definition of the universal enveloping algebra. In a basis-free form, you can take the tensor algebra $\bigotimes\mathfrak g$ of the vector space $\mathfrak g$ and factorize by the ideal generated by all elements of the form $X\otimes Y-Y\otimes X-[X,Y]$ for $X,Y\in \mathfrak g\subset\bigotimes \mathfrak g$. So you simply inforce the rule $XY-YX=[X,Y]$ in the quotient algebra. If you pick a basis $\{X_1,\dots,X_n\}$ for $\mathfrak g$ then you can identify $\bigotimes\mathfrak g$ with the algebra of non-commutative polynomials. The Lie bracket is then encoded via the so-called structure constants $C_{ij}^k$ defined by $[X_i,X_j]=\sum_kC_{ij}^kX_k$. Thus the realations you impose are exactly the relations $X\otimes Y-Y\otimes X-[X,Y]$ for the basis elements (which imply the corresponding relations for all elements of $\mathfrak g$ by bilinearity of the Lie bracket).
Concerning the second question, this is really a problem of bad notation. For an associative algebra $A$, let $A_L$ be the Lie algebra with the same underlying vector space and the commutator as a Lie bracket. Given a Lie algebra $\mathfrak g$, you should now look for homomorphisms $\mathfrak g\to A_L$ for unital associative algebras $A$. The universal enveloping algebra $\mathcal U(\mathfrak g)$ is the universal object for this quesiton, i.e. there is a homomorphism $i:\mathfrak g\to\mathcal U(\mathfrak g)_L$ such that any homomrophsim $\phi:\mathfrak g\to A_L$ can be written as $\tilde\phi\circ i$ for a homomorphism $\tilde\phi:\mathcal U(\mathfrak g)\to A$ of associative algebras. It then turns out that $i$ is injective, so you can also view $\mathcal U(\mathfrak g)$ as the "most general" associative algebra $A$ for which $A_L$ contains $\mathfrak g$ as a Lie subalgebra. 
