Left Bruck loop

Bruck-Bol definition
A loop is said to be a left Bruck loop or left K-loop or left gyrocommutative gyrogroup if it satisfies the following conditions:


 * It is a left Bol loop
 * It satisfies the automorphic inverse property (note that on account of its being a left Bol loop, it is already an inverse loop)

Gyrogroup definition
A gyrogroup $$(G,*)$$ with gyration map $$\operatorname{gyr}$$ is termed a left Bruck loop or left K-loop or left gyrocommutative gyrogroup if it satisfies the following identity called gyrocommutativity:

$$a * b = \operatorname{gyr}[a,b](b * a)$$

Weaker properties

 * Automorphic inverse property
 * Left Bol loop
 * Inverse loop
 * Power-associative loop