  The concept of weak Lie motion (weak Lie symmetry) is introduced through
${\cal{L}}_{\xi}{\cal{L}}_{\xi}g_{ab}=0,$
(${\cal{L}}_{\xi}{\cal{L}}_{\xi}f=0$). Applications are given which exhibit a
reduction of the usual symmetry, e.g., in the case of the the rotation group.
In this context, a particular generalization of Lie algebras is found
("extended Lie algebras") which turns out to be an involutive distribution or a
simple example for a tangent Lie algebroid. Riemannian and Lorentz metrics can
be introduced on such an algebroid through an extended Cartan-Killing form.
Transformation groups from non-relativistic mechanics and quantum mechanics
lead to such tangent Lie algebroids and to Lorentz geometries constructed on
them (1-dimensional gravitational fields).
