Lie algebras and cotriangular spaces

Let p (P,L) be a partial linear space in which any line contains three points and let K be a field. Then by LK(p) we denote the free K-algebra generated by the elements of P and subject to the relations xy = 0 if x and y are noncollinear elements from P and xy = z for any triple {x, y, z} ¿ L. We prove that the algebra LK(p) is a Lie algebra if and only if K is of even characteristic and p is a cotriangular space satisfying Pasch’s axiom. Moreover, if p is a cotriangular space satisfying Pasch’s axiom, then a connection between derivations of the Lie algebra LK(p) and geometric hyperplanes of p is used to determine the structure of the algebra of derivations of LK(p).