  A concept of the Moufang-Malt'tsev pair is elaborated. This concept is based on the generalized Maurer-Cartan equations of a local analytic Moufang loop. Triality can be seen as a fundamental property of such pairs. Based on triality, the Yamagutian is constructed. Properties of the Yamagutian are studied. 
Moufang symmetry II. Moufang-Mal'tsev pairs and triality