The universal enveloping algebra 
of a Lie algebra 
is the associative algebra
obtained from the tensor algebra 
, whose elements are finite linear combinations of tensors,
by imposing the relations ![xy-yx=[x,y]](/images/equations/UniversalEnvelopingAlgebra/Inline4.svg)
. More precisely,
 |
where 
is the two-sided ideal generated by ![x tensor y-y tensor x-[x,y]](/images/equations/UniversalEnvelopingAlgebra/Inline6.svg)
for 
. It is universal in the sense that every linear
transformation from 
to an associative algebra
that preserves the Lie bracket as a commutator
factors uniquely through 
. The Poincaré-Birkhoff-Witt
theorem describes a basis and gives an embedding
of 
into 
.
