**Definition 1.** A **Lie algebra** is an -algebra such that and

Hence provided is not a ring of characteristic Also, if is an -algebra, then we can define a Lie algebra whose product is defined by

Hence in particular, simply doubles the product of Recall that a derivation on an algebra satisfies

and

For let us define the **adjoint** of as defined by

**Proposition 2.** For is a derivation on

*Proof.* By tensorial properties of algebras we have

And

**Proposition 3.** Let be a commutative Lie algebra over a ring with Then If then is commutative, regardless of the ring.

*Proof.* If is commutative, then The second claim is trivial.

Let be a left ideal in and Then and hence So is also a right ideal. Hence we simply consider ideals of Lie algebras without reference to leftness or rightness.

**Definition 4.** Let be a submodule of such that for all derivations on then is called a **characteristic ideal** of

It’s easy to see that a characteristic ideal is actually an ideal in since we have

for all

**Proposition 5.** Let be an ideal of and be a characteristic ideal of then is an ideal of

*Proof.* If is a derivation on then since is a characteristic ideal. But multiplication by is a derivation on Hence for

For two ideals we have that and are ideals in and that and In the case of an -algebra , we will let denote the submodule generated by elements This is in fact an ideal in a Lie algebra for if is a derivation, then

Hence it is stable under adjoint action.

**Definition 6.** The ideal is called the** derived ideal** of Inductively we can define

The sequence is called the **derived series** of Let us define and

The sequence is called the **lower central series** of

**Definition 7.** Let be a Lie algebra, be its tensor algebra (with interpreted as a module), and be the ideal generated by elements of the form Then is called the **enveloping algebra** of

**Theorem 8.** The enveloping algebra satisfies the universal property that if is an algebra homomorphism for some associative unital algebra and is the canonical inclusion, then there exists a unique morphism and such that

**Proposition 9.** If and are Lie algebras, then

Let us define with defined as above, and It follows that is a filtered algebra, which we will call the **associated filtered algebra** to

**Theorem 10 (Poincare-Birkhoff-Witt)** Let be the associated filtered algebra to and be the symmetric algebra of as a module. Then if is free,

It follows that if is free, then the canonical homomorphism is injective.

[1] Bourbaki, Nicholas. *Lie Groups and Lie Algebras, Chapters 1-3*. Springer-Verlag. 1971.