I am working on a research paper where I need to investigate conditions for the existence of probability distributions satisfying certain characteristics. I have already asked a related question (here), whose answers allowed me to frame better in my mind the problems I'm facing. In what follows, I report a close, although different, question. I will highlight the key differences below.

Consider a $6\times 1$ random vector $$ \eta\equiv (\eta_1,\eta_2,..., \eta_6) $$ satisfying the following property (hereafter, called Property 1):

**Property 1:** $$
\begin{pmatrix}
\eta_1\\
\eta_2\\
\eta_3
\end{pmatrix} \sim \begin{pmatrix}
\eta_4\\
-\eta_2\\
\eta_5
\end{pmatrix} \sim \begin{pmatrix}
\eta_6\\
-\eta_3\\
-\eta_5
\end{pmatrix} \sim \begin{pmatrix}
-\eta_1\\
-\eta_4\\
-\eta_6
\end{pmatrix} \sim G
$$
where "$\sim$" denotes "distributed as" and $G$ is an absolutely continuous distribution with full support on $\mathbb{R}^3$.

**Question A:** Let $\mathcal{G}$ denote the family of absolutely continuous distribution with full support on $\mathbb{R}^3$ and whose marginals are *symmetric around zero and identical*. For each $G\in \mathcal{G}$, does there exists a vector $\eta$ satisfying Property 1?

**Question B:** Let $\epsilon$ be a $4\times 1$ random vector
$$
\epsilon\equiv \begin{pmatrix}
\epsilon_1\\
\epsilon_2\\
\epsilon_3\\
\epsilon_0\\
\end{pmatrix} $$

For each $(G,\eta)$ satisfying Property 1, does there exist $\epsilon$ satisfying Property 2 described below?

**Property 2:**
$$
\begin{pmatrix}
1 & 0 & 0 & -1\\
1 & -1 & 0 & 0\\
1 & 0 & -1 & 0\\
0 & 1 & 0 & -1\\
0 & 1 & -1 & 0\\
0 & 0 & 1 & -1\\
\end{pmatrix}*\epsilon=\begin{pmatrix}
\eta_1\\
\eta_2\\
\eta_3\\
\eta_4\\
\eta_5\\
\eta_6
\end{pmatrix}
$$
and the distribution $F$ of $\epsilon$ is absolutely continuous with full support on $\mathbb{R}^4$?

**My thoughts:**

I believe that the answer to Question A is "Yes": any distribution in $\mathcal{G}$ satisfies Property 1. Certainly, there exist distributions outside $\mathcal{G}$ that can also satisfy Property 1.

I believe that the answer to Question B is "Yes" as well. However, I'm not 100% sure and I would appreciate your help. The answers here suggest that: if $F$ is absolutely continuous with full support on $\mathbb{R}^4$, then $G$ is absolutely continuous with full support on $\mathbb{R}^3$. Here, however, I'm asking something different: if $G$ is absolutely continuous with full support on $\mathbb{R}^3$, can we always find a distribution $F$ for $\epsilon$ that is absolutely continuous with full support on $\mathbb{R}^4$?