Proving $\mathcal{D}-class$ of semigroup.

Proving $\mathcal{D}-class$ of semigroup.



The problem is in LEMMA 8.35, how can I choose $b$ and $c$. And $b$ and
$c$ should be idempotent?
I know that if $a \in D_{e}$, then there exist $x,y \in S^{1}$such that
$x \in R_{e} \cap L_{a}$ and $y \in R_{a} \cap L_{e}$. Thank you.