Gyrogroups were discovered by Ungar in modelling the Einstein velocity addition rule in relativity. Have they been shown to be useful elsewhere in mathematics (or mathematical physics)?
Gyrogroups have been used in the theory of signal processing:
Gyrocommutative gyrogroups were independently introduced by Bruck, and are now called Bruck loops or K-loops in the quasigroup literature. He presumably invented them for some reason, but I couldn't find out a quick answer why.
Bruck loops do arise naturally in the study of sharply 3-transitive groups. You can associate a generalized ring, known as a neardomain, to every sharply 3-transitive group. The addition operation is then a Bruck loop. Kiechle has some slides, Is every Neardomain a Nearfield?, that mention the connection. In the finite case, the additive loop of a neardomain is forced to be an abelian group operation, but recently Rips, Segev, and Tent have shown that there are infinitely many infinite examples. See Tent - Infinite sharply multiply transitive groups for more.