Consider we have three random variables $x_1, x_2, x_3$ that are jointly multivariate normal. Assume also a bijective function $f:\mathbb{R}^2\rightarrow \mathbb{R}^2$.
How may I compute the joint density of $(y_1, y_2, y_3, y_4)$ given that: \begin{align} (y_1, y_2) &= f(x_1, x_2) \,\, \text{ and}\\ (y_3, y_4) &= f(x_2, x_3) \end{align}
First thought was that I could define joint density of $(x_1, x_2, x_2, x_3)$ and then use the formula for transforming probability densities as per usual with the Jacobian etc,,, but this doesn't seem appropriate as the above vector has $x_2$ repeated and, I believe, it's not possible to define a density for a random variable with itself given it would have measure zero?
Second idea was to instead to consider the multivariate normal distribution of the three random variables $(x_1, x_2, x_3)$ which is clearly well defined, but in that case I would not be able to compute the Jacobian as I would have to define an appropriate new function $\bar{f}:\mathbb{R}^3 \rightarrow \mathbb{R}^4$ and such jacobian would not be square. Does the joint density that I'm looking for exist and how can I derive it?