evolution - $F_{ST}$ and the genetic variance in metapopulations

From this video (21'15''), the speaker gives the following formulae in order to calculate the between and among populations genetic variance from the $F_{ST}$:

$$V_{Among Pop} = 2 F_{ST}V_G$$

$$V_{Within Pop} = (1-F_{ST})V_G$$

, where $V_G$ is the total genetic variance of this population if it was well mixing. $V_{Among Pop}$ is the variance among populations and $V_{Within Pop}$ is the variance within populations.

In this same video, the speaker defines $F_{ST}$ as:

$$F_{ST} = \frac{\text{Var}(p)}{\bar p (1-\bar p)}$$

, where $p$ is a vector of frequencies of a given allele and $\bar p$ and $\text{Var}(p)$ are the mean and variance of this vector.

For example, consider a metapopulation made of 4 subpopulations. The allele frequencies in these 4 subpopulations are p=[0.2, 0.5, 0.8, 0.3]. $\bar p$ is the mean of $p$ ($\bar p = 0.45$) and $\text{Var}(p)$ is the variance of $p$ ($\text{Var}(p )=0.07 \space$).

Can you help me to make sense of the formulas for $V_{Among Pop}$ and $V_{Within Pop}$?

Can you prove these formulas? I would expect that $V_{Among Pop} + V_{Within Pop} = V_G$ but it doesn't! Maybe the issue has to do with the fact that $V_{Within Pop}$ as defined above fit to haploid population and not diploid populations. Then, I would expect that for diploid populations $V_{Within Pop}$ becomes $(1-2F_{ST})V_G$. Is that correct? Just in order to make it more general, how do you extrapolate these definitions for a tetraploid population? Also maybe I misunderstand the meaning of $V_G$. Is the variance within population the sum of the within populations variance or is it the average (or something else)? Thanks for your help!