The symbols for variance and covariance

Question

I learned to denote the variance of $x$ as $\sigma_x^2$, and the covariance of $x$ and $y$ as $\sigma_{x, y}$.

The covariance of $x$ and $x$ is then $\sigma_{x, x}$, but because that it just the variance of $x$, I am told that it must be written $\sigma_x^2$, not $\sigma_{x, x}$. Why?

For example, I see equations like this:

$\sigma_P^2 =\sum_{j=1}^N{X_j^2\sigma_j^2} + \sum_{j=1}^N{\sum_{\substack{k=1 \\ k \neq j}}^N{X_jX_k\sigma_{jk}}}$

Why not just:

$\sigma_P^2 = \sum_{j=1}^N{\sum_{k=1}^N{X_jX_k\sigma_{jk}}}$

?

Answer 1

Both notations are fine. I'm not sure why someone would care that much.