Associativity of Relativistic Oblique Velocity Addition

I've encountered some information in the Wikipedia page on Lorentz transformation (https://en.m.wikipedia.org/wiki/Lorentz_transformation) that I am having difficulty reconciling with other information that I found on the page (https://en.m.wikipedia.org/wiki/Wigner_rotation) on Wigner rotation. In the Lorentz Transformation page, it says that the oblique (non-collinear) Lorentz transformation is given by a matrix $$\begin{pmatrix}\gamma&-\gamma\beta n_x&-\gamma\beta n_y&-\gamma\beta n_z\\-\gamma\beta n_x&1+(\gamma-1)n_x^2&(\gamma-1)n_x n_y&(\gamma-1)n_z n_x\\-\gamma\beta n_y&(\gamma-1)n_x n_y&1+(\gamma-1)n_y^2&(\gamma-1)n_y n_z\\-\gamma\beta n_z&(\gamma-1)n_z n_x&(\gamma-1)n_y n_z2&1+(\gamma-1)n_z^2\end{pmatrix}$$ with the $$n_q$$ ($$q\in\{x,y,z\}$$) being the components of the unit vector along the direction-of-motion of the moving frame. And in the Wigner Rotation page, there is given the formula for oblique velocity addition that is derived from successive application of two such transformations $$\overrightarrow{\beta}_u\oplus\overrightarrow{\beta}_v=\frac{1}{1+\overrightarrow{\beta}_u.\overrightarrow{\beta}_v}\left[\left(1+\frac{\gamma_u\overrightarrow{\beta}_u.\overrightarrow{\beta}_v}{1+\gamma_u}\right)\overrightarrow{\beta}_u+\frac{\overline{\beta}_v}{\gamma_u}\right]$$$$\gamma_{\overrightarrow{\beta}_u\oplus\overrightarrow{\beta}_v}=\gamma_u\gamma_v(1+\overrightarrow{\beta}_u.\overrightarrow{\beta}_v) .$$ But then it goes on to say that in addition to being non-commutative (which I can well-accomodate), this recipe is also non-associative!:

"Although velocity addition is nonlinear, non-associative, and non-commutative, the result of the operation correctly obtains a velocity with a magnitude less than $$c$$.".

Now since the addition of velocities is derived from the application of successive transformations taking the mathematical form of matrices, which are certainly associative, how then does the non -associativity of the addition of velocities come-about? For three consecutive Lorentz transformations would have result not contingent upon the association of the matrices - and the resultant velocity is derived from this product of three matrices ... so how can it come-about that the velocity addition be non-associative?

Say you have four frames of reference labelled 0 to 3, the zeroth frame being the 'base' one, if you will; and let $$Q_{ab}$$ be the value of $$Q$$ - with Q being $$\gamma$$ or $$\beta$$ (or any other quantity, for that matter) - of frame $$b$$ relative to frame $$a$$, with $$a\in\{0\dots3\}$$ & $$b\in\{0\dots3\}$$. This way, $$\beta_{01}$$, $$\beta_{12}$$ & $$\beta_{23}$$ - & therefore $$\gamma_{01}$$, $$\gamma_{12}$$ & $$\gamma_{23}$$ are the givens. If you apply the formula first to find the velocity of frame 3 relative to frame 1, you would use the addition-of-velocity formula as given with $$\gamma_u=\gamma_{12}$$, $$\beta_u=\beta_{12}$$, & $$\beta_v=\beta_{23}$$, and the result would be $$\beta_{13}$$. Then to find the velocity of frame 3 relative to frame 0, you would simply iterate the formula with $$\gamma_u=\gamma_{12}$$, $$\beta_u=\beta_{12}$$, & $$\beta_v=\beta_{23}$$, and the result would be $$\beta_{03}$$ - the result sought. However, associating the calculation (the calculation that is! ... not the formula itself, as will shortly be apparent), you would first calculate $$\beta_{02}$$ by substituting $$\gamma_u=\gamma_{01}$$, $$\beta_u=\beta_{01}$$, & $$\beta_v=\beta_{12}$$. But then to calculate $$\beta_{03}$$ you would need to substitute $$\gamma_u=\gamma_{02}$$, $$\beta_u=\beta_{02}$$, & $$\beta_v=\beta_{23}$$. Note that in this association of the calculation $$\gamma_{02}$$ appears - which must be calculated apart using the ancilliary formula for composition of the Lorentz factor.