Introduction to Moufang symmetry

In the present paper, the differential equations of a local analytical (l.a.) Moufang loop are established which turn out to be a natural generalization of the Lie and Maurer-Cartan equations from the Lie groups theory. By using the triality-property of the generalized Maurer-Cartan equations the Sagle-Yamaguti identity (equivalent to the Malcev one) is proved to hold in the tengent space Te of the identity element e of the l.a. Moufang loop. Also, the commutation relations of the most important Lie algebras associated with a given l.a. Moufang loop are established