The abelian group of vectors in a vector space can also be given a new structure by defining multiplication between vectors to get another vector. The Cartesian inspiration here might be considered to be the vector **cross product** (AKA vector product or outer product) in 3 dimensions.

**Algebra**: defines a bilinear product distributive over vector addition (no commutativity, associativity, or identity required)**Associative algebra**: associative product; turns the abelian group of vectors into a ring; can always be naturally extended to include a multiplicative identity**Lie algebra**: product denoted \({\left[u,v\right]}\); satisfies two other attributes of the cross product, anti-commutativity \({\left[u,v\right]=-\left[v,u\right]}\) and the**Jacobi identity**\({\left[\left[u,v\right],w\right]+\left[\left[w,u\right],v\right]+\left[\left[v,w\right],u\right]=0}\)

For example, the Cartesian vectors under the cross product are a non-associative Lie algebra, while the real \({n\times n}\) matrices under matrix multiplication are an associative algebra.

Δ Note that “algebra” is sometimes defined to include associativity and/or an identity. |

Δ If the scalars of a Lie algebra are a field of characteristic 2, then we no longer have \({\left[u,v\right]=-\left[v,u\right]\Rightarrow\left[v,v\right]=0}\), and the latter is imposed as a separate requirement in the definition. |

In a Lie algebra (pronounced “lee”), the product is called the **Lie bracket**, and the notation \({\left[u,v\right]}\) in place of of \({uv}\) reflects the close relationship between Lie algebras and associative algebras: every associative algebra can be turned into a Lie algebra by defining the Lie bracket to be \({\left[u,v\right]\equiv uv-vu}\). In these cases the Lie bracket is called the **Lie commutator**. The **Poincaré-Birkhoff-Witt theorem** provides a converse to this: that every Lie algebra is isomorphic to a subalgebra of an associative algebra called the **universal enveloping algebra** under the Lie commutator. An **abelian algebra** has \({uv=vu}\); thus an **abelian Lie algebra** (AKA commutative Lie algebra) has \({\left[u,v\right]=\left[v,u\right]\Rightarrow\left[u,v\right]=0}\). Note that an associative Lie algebra is not necessarily abelian, but does satisfy \({\left[\left[u,v\right],w\right]=0}\) via the Jacobi identity.

Any algebra over a field is completely determined by specifying scalars called **structure coefficients** (AKA structure constants), defined in a given basis as follows:

\(\displaystyle e_{\mu}e_{\nu}=c^{\rho}{}_{\mu\nu}e_{\rho} \)

However, different structure coefficients may define isomorphic algebras.