  By a completely inverse $AG^{**}$-groupoid we mean an inverse
$AG^{**}$-groupoid $A$ satisfying the identity $xx^{-1}=x^{-1}x$, where
$x^{-1}$ denotes a unique element of $A$ such that $x=(xx^{-1})x$ and
$x^{-1}=(x^{-1}x)x^{-1}.$ We show that the set of all idempotents of such
groupoid forms a semilattice and the Green's relations $\mathcal{H,L, R,D}$ and
$\mathcal{J}$ coincide on $A$. The main result of this note says that any
completely inverse $AG^{**}$-groupoid meets the famous Lallement's Lemma for
regular semigroups. Finally, we show that the Green's relation $\mathcal{H}$ is
both the least semilattice congruence and the maximum idempotent-separating
congruence on any completely inverse $AG^{**}$-groupoid.
