Vectors in Three Dimensional Space with Cross Product forms Lie Algebra

Theorem
Let $S$ be the set of vectors in $3$ dimensional Euclidean space.

Let $\times$ denote the vector cross product on $S$.

Then $\struct {S, \times}$ is a Lie algebra.

Proof
By definition of Lie algebra, it suffices to prove two properties:


 * $(1): \forall a \in S: a \times a = 0$


 * $(2): \forall a, b, c \in S: a \times \paren {b \times c} + b \times \paren {c \times a} + c \times \paren {a \times b} = 0$

Proof of $(1)$
Cross Product of Vector with Itself is Zero

Proof of $(2)$
Vector Cross Product satisfies Jacobi Identity

Both properties hold, and the result follows.