I'm working with Lorentz transformations, and one of the problems I encountered needed a derivation of the velocity of a particle $u$ in a stationary reference frame S, in the variables $u'$, which is the velocity measured of the particle of an observer moving with another reference frame S', that's moving with a velocity $v$ to the right.
Either way, the problem is one dimensional with respect to spatial dimensions, meaning, we're only dealing with $x$-coordinates.
I managed to find it to be $$u = \frac{u' + v}{1 + \frac{u' v}{c^2}}.$$ Then I encounted a question where I needed to argue why velocity addition under the Lorentz transformation is commutative, and associative, in the case when we have a boost in only one direction. I did some research and found out that this isn't always the case for when we have boosts in multiple directions (since a boost can be described my a matrix, two boosts after one another (that're not in the same direction) is a product of two matricies, for which the order matters).
How can one intuitively explain why it's associative and commutative under my given conditions?