Given a set ÎŁ of equations, the free-algebra functor FÎŁâ
associates to each set X of variables the free algebra FÎŁâ(X) over
X. Extending the notion of \emph{derivative} ÎŁâČ for an arbitrary set
ÎŁ of equations, originally defined by Dent, Kearnes, and Szendrei, we
show that FÎŁâ preserves preimages if and only if ÎŁâąÎŁâČ, i.e. ÎŁ derives its derivative ÎŁâČ. If FÎŁâ weakly
preserves kernel pairs, then every equation p(x,x,y)=q(x,y,y) gives rise to a
term s(x,y,z,u) such that p(x,y,z)=s(x,y,z,z) and q(x,y,z)=s(x,x,y,z). In
this case n-permutable varieties must already be permutable, i.e. Mal'cev.
Conversely, if ÎŁ defines a Mal'cev variety, then FÎŁâ weakly
preserves kernel pairs. As a tool, we prove that arbitrary Setâendofunctors
F weakly preserve kernel pairs if and only if they weakly preserve pullbacks
of epis
We present all real solvable algebraically rigid Lie algebras of dimension lower or equal than eight. We point out the differences that distinguish the real and complex classification of solvable rigid Lie algebra