How to calculate a cross-product like R^2? $\lbrace Y_1, Y_2, \boldsymbol{X} \rbrace$ are jointly normally distributed (it is not essential to assume normality, I think).  Let $\Sigma_{X}$ be the variance-covariance matrix of $\boldsymbol{X}.$  Let $\Sigma_{1X}$ and $\Sigma_{2X}$ be covariance vectors of $Y_1$ and $Y_2$ with $\boldsymbol{X},$ respectively. Let $\sigma_1$ and $\sigma_2$ be the standard deviations of $Y_1$ and $Y_2.$  Suppose I have a sample drawn from this multivariate distribution: $\lbrace y_{1i}, y_{2i}, \boldsymbol{x}_i \rbrace, i=1, \ldots, N$.  I can estimate $S_1 = (1/\sigma_1^2) \Sigma_{1X} \Sigma_{X}^{-1} \Sigma_{1X}$ as the $R^2$ from the linear regression of $y_1$ versus $\boldsymbol{x}$. Similarly I can estimate $S_2 = (1/\sigma_2^2) \Sigma_{2X} \Sigma_{X}^{-1} \Sigma_{2X}$ as the $R^2$ from the linear regression of $y_2$ versus $\boldsymbol{x}$.
Now here is my question:  How can I estimate the quantity $T = (1/\sigma_1 \sigma_2)\Sigma_{2X} \Sigma_{X}^{-1} \Sigma_{1X}$ without directly computing the covariance matrices? Is there a good interpretation of $T?$
I tried combining the $R^2$ estimates from the 3 regression models ($y_1 \sim \boldsymbol{x}$, $y_2 \sim \boldsymbol{x}$, and $y_2-y_1 \sim \boldsymbol{x}$) to estimate $T$, but it is not working.

I was able to get an answer as follows:
$$T = \left[ S_1 \sigma_1^2 + S_2 \sigma_2^2 - S_{12}(\sigma_1^2 + \sigma_2^2 - 2\rho \sigma_1 \sigma_2) \right] /(2\sigma_1 \sigma_2),$$ where $\rho$ is the correlation between $Y_1$ and $Y_2.$  I would still appreciate other/better ways of estimating this, as well as how to interpret this quantity $T.$
