Are these properties sufficient to characterize the cross product in three dimensions?

Question

Given that $\times$ is an operation that takes two three-dimensional vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ and outputs one three-dimensional vector $\boldsymbol{a} \times \boldsymbol{b}$, are these properties sufficient to prove that it is the cross product?

1. Magnitude

$\left \| \boldsymbol{a}\times \boldsymbol{b} \right \|=\left \| \boldsymbol{a} \right \|\left \| \boldsymbol{b} \right \|\sin (\widehat{\boldsymbol{a};\boldsymbol{b}})$

2. Right hand rule

$\mathbf{\hat{i}} \times\mathbf{\hat{j}}=\mathbf{\hat{k}}$

3. Continuity

$\lim_{\Delta \boldsymbol{a}\rightarrow \mathbf{0}}(\boldsymbol{a}+\Delta \boldsymbol{a}) \times \boldsymbol{b} = \boldsymbol{a} \times \boldsymbol{b}$

$\lim_{\Delta \boldsymbol{b}\rightarrow \mathbf{0}} \boldsymbol{a} \times (\boldsymbol{b}+\Delta \boldsymbol{b}) = \boldsymbol{a} \times \boldsymbol{b}$

Answer 1

No, all of these axioms are satisfied by the function $\| a \| \| b \| (\sin \theta) \hat{\bf{k}}$. It's a bad idea to try to characterize the cross product without including bilinearity!

The following axioms, on the other hand, do uniquely characterize the cross product:

1. Bilinearity: $(ax + by) \times z = a x \times z + b y \times z$ and the same in the second variable.

2. Antisymmetry: $x \times y = - y \times x$.

3. $x \times y$ is orthogonal to the plane spanned by $x$ and $y$, or equivalently $x \cdot (x \times y) = 0$ (which implies the same for $y$ by antisymmetry).

4. Right hand rule: $\hat{\bf{i}} \times \hat{\bf{j}} = \hat{\bf{k}}$.

Note that continuity follows automatically from bilinearity. This is a nice exercise; the idea is that the first three axioms imply that the triple product $x \cdot (y \times z)$ is trilinear and alternating (meaning it is equal to zero if any two of its inputs are equal) which means it must be a scalar multiple of the determinant of the matrix with columns $x, y, z$, and the fourth axiom fixes which multiple it is.

Alternatively, the following axioms also uniquely characterize the cross product:

1. Bilinearity again.
2. Rotation equivariance: if $R \in SO(3)$ is a rotation, then $(Rx) \times (Ry) = Rz$.
3. Right hand rule again.

This is because rotation equivariance together with the right hand rule determines the cross product of any two basis vectors and then we're done by bilinearity.

In the background here are some abstract facts about the representation theory of the Lie group $SO(3)$ and its Lie algebra $\mathfrak{so}(3)$ although none of that is strictly speaking necessary, it just helps organize what's going on.