I'm using Ungar's terminology and notations. In the open unit ball of $\mathbb{R}^n$, let $GB(A_1, \ldots, A_N; m_1, \ldots, m_N)$ be the gyrobarycenter of the points $(A_1, \ldots, A_N)$ with gyrobarycentric coordinates $(m_1, \ldots, m_N)$.

I numerically found that (for $n=2$, I have not checked for other values of $n$) $$ GB(A_1,A_2,A_3;1,1,1) = GB\bigl(GB(A_1,A_2;1,1), A_3; 2\gamma_{(-A_1)\oplus A_2}, 1\bigr) $$ and $$ GB(A_1,A_2,A_3;2,2,1) = GB\bigl(GB(A_1,A_2;1,1), A_3; 4\gamma_{(-A_1)\oplus A_2}, 1\bigr). $$ I also found that there is $p$ not depending on $A_3$ such that $$ GB(A_1,A_2,A_3;2,1,1) = GB\bigl(GB(A_1,A_2;2,1), A_3; p, 1\bigr) $$ but I didn't find the expression of $p$.

So it looks like there is an associativity property of the gyrobarycenter. What is this property? There is nothing about that in Ungar's books.

EDIT

The gyrobarycenter (I'm interested in the Möbius gyrovector space) is defined as $$ GB(A_1, \ldots, A_N; m_1, \ldots, m_N) = \frac{1}{2} \otimes \frac{\sum m_k \gamma^2_{A_k} A_k} {\sum m_k \Bigl(\gamma^2_{A_k}-\frac{1}{2}\Bigr)}. $$ The $\gamma$ factor of a point $X$ is defined by $$ \gamma_X = \frac{1}{\sqrt{1-\Vert X \Vert^2}}. $$ The scalar multiplication $\otimes$ is defined at page 10 of this paper, and the gyroaddition $\oplus$ at page 9.