There's a fascinating history of how out of the four founding fathers of modern multi-dimensional mathematics, three preferred geometric algebra, and only one preferred vector and matrix algebra. The three all left academia or died young, the one dissenter lived to a ripe old age and was a vocal proponent of his preferred approach -- because it's more useful for statistics. He wasn't a physicist.

All of these weird "triple products", limitations to 3D space, etc... vanish in geometric algebra. You can use the exact same formulas in 2D space, 3D space, 4D spacetime, or 18 dimensional spaces with degenerate dimensions if you please.

There's this obstinate refusal to just admit that the maths that's ideally suited to solving statistical problems may not be ideal for physics, robotics, or optics. Instead, physicists insist on re-inventing the good stuff over and over, badly, with different names, each time.

All of the following are just geometric algebra in disguise, or various "subsets" of a geometric algebra, or a geometric algebra operation that got renamed:

    - Cross products
    - Triple product
    - Exterior algebra
    - Complex numbers
    - Quaternions
    - Octonions
    - Pauli matrices
    - Gamma matrices

And on, and on, and on.

As a random example: the way we represent 3D rotations using a vector in 3D doesn't work in 2D, because a 2D rotation vector would point out of the plane. It also doesn't work in 4D or higher dimensions. It just happens to "work" in 3D not because 3D is natural, preferred, or special, but because the broken maths happens to not be completely broken in one case due to a simple coincidence.

In geometric algebra, instead of using a vector, a bivector is used, which is like a surface patch. Notice that you can have a surface in 2D, so rotations in GA work in 2D. You can also have a surface in 3D, so rotations work in 3D. And in 4D (including both 4D space and 3+1D space-time!), and 5D... and all of rest, with the same formula!

PS: Geometric algebra also prevents gimbal lock, doesn't store redundant values, and has better numerical precision. Its transformations can be interpolated unlike (famously!) matrix transformations, which can't. In robotics or computer graphics this means everyone uses quaternions instead, which are... drumroll... the "even subset" of a geometric algebra.