Why is compound symmetry sufficient to establish sphericity?

At this website it is stated that "If compound symmetry is met then sphericity is also met."

Is it impossible to construct a data set that has perfect compound symmetry but not perfect sphericity? Why/why not? It's not intuitively obvious to me because at face value compound symmetry (variances across conditions approximately equal, and covariances between pairs of conditions also approximately equal) and sphericity (approximately equal variances of the differences between data from the same participant at different times) seem like quite different things.

I'm thinking about the context of repeated-measures ANOVA.

You can check sphericity using the covariance matrix: if $\sigma^2_i+\sigma^2_{j}-2\sigma_{ij}=k$ (the same $k$) for all $i,j$ then sphericity holds.
If compound symmetry holds, then all the variances and all the covariances are equal, so that requirement is met, e.g. $\sigma^2_1+\sigma^2_2-2\sigma_{12}=\sigma^2_1+\sigma^2_3-2\sigma_{13}$ because $\sigma^2_1=\sigma^2_2=\sigma^2_3$ and $\sigma_{12}=\sigma_{13}$.