  In this paper we study the classes of superconnected and superfaithful left
quasigroups, that are relevant in the study of Mal'cev varieties of left
quasigroups \cite{Maltsev_paper}. Then we focus on quandles and in particular
to the involutory ones. We extend the main result of
\cite{involutive_quandles_russo} to the infinite case and we offer a
characterization of several classes of involutory quandles in terms of the
properties of the canonical generators of the displacement group, improving the
main results of \cite{Nobu}.
