# Non-associative Lie algebra

#### Lapidus

How come that a Lie algebra is defined as being non-associative and at same time it describes a Lie group locally? I wonder because groups are, again by definition, asscioative.

thanks

#### Dickfore

Let

$$a \otimes b \equiv [a, b] = a b - b a$$

Then:

$$a \otimes (b \otimes c) = [a,[b,c]] = [a, b c] - [a, c b] = a b c - b c a - a c b + c b a$$

But:

$$(a \otimes b) \otimes c = [[a, b], c] = [a b, c] - [b a, c] = a b c - c a b - b a c + c b a$$

These two expressions are not the same. Thus, the operation $\otimes$ is not associative. But, the Lie algebra is precisely defined through such an operation.