Einstein Gyrogroup as a B-loop

Using the Clifford algebra formalism, we give an algebraic proof that the open unit ball B={v∈Rn:‖v‖<1} of RnRn equipped with Einstein addition ⊕E forms a B-loop or, equivalently, a uniquely 2-divisible gyrocommutative gyrogroup. We obtain a compact formula for Einstein addition in terms of Möbius addition. We then give a characterization of associativity and commutativity of vectors in BB with respect to Einstein addition.