I came up with a proof by the method of characteristic function, but it might be too simple to be true. So I am unsure about that.
The problem is as follows: suppose four random variables X, Y, T and Z are mutually independent, then the random vector (X+Y, X+T) and the r.v. Z are independent?
Here is my argument:
For any $(t_1,t_2,s) \in \mathbb{R}^3$,
\begin{align} \phi_{X+Y,X+T,Z}(t_1,t_2,s)&= \mathbb{E}[e^{it_1(X+Y)+it_2(X+T)+is Z }] \\ &=\mathbb{E}[e^{i(t_1+t_2)X+it_1 Y+it_2 T + is Z}] \\ &=\mathbb{E}[e^{i(t_1+t_2)X+it_1 Y+it_2 T}]\mathbb{E}[e^{is Z}] \qquad\text{(by the mutual independence)}\\ &=\phi_{X+Y,X+T}(t_1,t_2)\cdot\phi_Z(s) \end{align}
So $(X+Y, X+T)$ and $Z$ are independent.
Did I miss some points here?