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

$\Box$

Proof of $(2)$

Vector Cross Product satisfies Jacobi Identity

$\Box$

Both properties hold, and the result follows.

$\blacksquare$

Sources

• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.26$: Extensions of the Complex Number System. Algebras, Quaternions, and Lagrange's Four Squares Theorem