Consider a $6\times 1$ continuous random vector $$ \eta\equiv (\eta_1,\eta_2,..., \eta_6) $$ satisfying the following property: $$ \underbrace{\begin{pmatrix} \eta_1\\ \eta_2\\ \eta_3 \end{pmatrix}}_{\equiv x_1} \sim \underbrace{\begin{pmatrix} \eta_1\\ \eta_4\\ \eta_5 \end{pmatrix}}_{\equiv x_2} \sim \underbrace{\begin{pmatrix} \eta_2\\ \eta_4\\ \eta_6 \end{pmatrix}}_{\equiv x_3} \sim \underbrace{\begin{pmatrix} \eta_3\\ \eta_5\\ \eta_6 \end{pmatrix}}_{\equiv x_4} \sim G $$ where ``$\sim$'' denotes "distributed as" and $G$ is some distribution with support $\subseteq \mathbb{R}^3$.

**Question:** I am looking for necessary and sufficient conditions on the support of $G$ such that there exists a $4\times 1$ random vector $\epsilon \equiv (\epsilon_0, \epsilon_1,\epsilon_2,\epsilon_3)$ having an absolutely continuous distribution with full support on $\mathbb{R}^4$ and such that
$$ (*) \quad \quad
\begin{aligned}
\eta_1= \epsilon_1-\epsilon_0\\
\eta_2= \epsilon_2-\epsilon_0\\
\eta_3= \epsilon_3-\epsilon_0\\
\eta_4= \epsilon_1-\epsilon_2\\
\eta_5= \epsilon_1-\epsilon_3\\
\eta_6= \epsilon_2-\epsilon_3\\
\end{aligned}
$$

In particular, I would like to understand the following:

A) if the distribution of $\epsilon$ has full support on $\mathbb{R}^4$, then $G$ CANNOT have full support on $\mathbb{R}^3$. Is this correct? Why?

B) if A is correct, then there should be certain boxes in $\mathbb{R}^3$ where $G$ is zero. Can we characterise those boxes?