Nilpotent Sabinin algebras

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In this note we establish several basic properties of nilpotent Sabinin algebras. Namely, we show that nilpotent Sabinin algebras (1) can be integrated to produce nilpotent loops, (2) satisfy an analogue of the Ado theorem, (3) have nilpotent Lie envelopes. We also give a new set of axioms for Sabinin algebras. These axioms reflect the fact that a complementary subspace to a Lie subalgebra in a Lie algebra is a Sabinin algebra. Finally, we note that the non-associative version of the Jennings theorem produces a version of Ado theorem for loops whose commutator-associator filtration is of finite length.