Alternatively, a right or left Bol loop is Moufang if and only if it satisfies the flexible identity a(ba) = (ab)a .
A Bol loop where the aforementioned two-sided inverse satisfies the automorphic inverse property, (ab)−1 = a−1 b−1 for all a,b in L, is known as a (left or right) Bruck loop or K-loop (named for the American mathematician Richard Bruck).
Left Bruck loops are equivalent to Ungar's (2002) gyrocommutative gyrogroups, even though the two structures are defined differently.
It is generally not true that the matrix product AB of matrices A, B in L is Hermitian, let alone positive definite.
An explicit formula for * is given by A * B = (A B2 A)1/2, where the superscript 1/2 indicates the unique positive definite Hermitian square root.
A (left) Bol algebra is a vector space equipped with a binary operation