I'm working out a derivation of a posterior distribution that's bivariate normal, and I've gotten to the point where I know it's proportional to:
$$\exp\left[- \frac { x_1^2 (\delta^2 + \sigma_2^2) + x_2^2 (\delta^2 + \sigma_1^2) - 2 \delta^2 x_1 x_2 - 2\sigma_2^2 \nu x_1 - 2 \sigma_1^2 \nu x_2 }{2 \left ( \sigma_1^2 \sigma_2^2 + \sigma_1^2 \delta^2 + \sigma_2^2 \delta^2 \right)} \right]$$
In the above, the variable vector is $(x_1, x_2)$. It seems to me like it should be straightforward, then, to find the mean vector $\vec \mu$ and the variance-covariance matrix $\mathbf \Sigma$, but I don't know of any way to do it other than literally doing algebraic manipulation and creating a system of equations for the five variables I need to find ($\mu_{x_1}, \mu_{x_2}, \sigma_{x_1}^2, \sigma_{x_2}^2, \rho$) and then solving it, but a preliminary look into that seems to be a huge mess.
Is there any other way to find the values of my coefficients?
Compare the expression with the simplified version of the bivariate normal (see this for example, page $3$).