Lie algebra

Remarks

Example: $\R^3$ with the cross product $[x, y] = x \times y$ is a real Lie algebra. The cross product is bilinear and $x \times x = 0$. The Jacobi identity follows from the rule $x \times (y \times z) = (x \cdot z)\, y - (x \cdot y)\, z$: summing over the three cyclic permutations, the six terms cancel in pairs.

Non-example: $\R$ with the bracket $[x, y] = xy$ is bilinear, but it fails clause 1, since $[1, 1] = 1 \ne 0$.

Clause 1 and bilinearity give anticommutativity, $[x, y] = -[y, x]$, by expanding $[x + y, x + y] = 0$. If the characteristic of $k$ is not $2$, anticommutativity implies clause 1. The same definition with a commutative ring in place of $k$ and a module in place of the vector space gives Lie algebras over a commutative ring. Lie algebras are usually written in Fraktur letters such as $\mathfrak g$, $\mathfrak h$ and $\mathfrak{sl}_2$.