So far nobody is actually giving properties, but just examples. I'll continue that theme.
When Gauss defined composition of quadratic forms, on the level of quadratic forms what he defined was not really a law of composition but a ternary relation (three quadratic forms $Q_1$, $Q_2$, and $Q_3$ are "in composition" if $Q_1(x,y)Q_2(x',y') = Q_3(B,B')$ where $B$ and $B'$ are linear in $xx', xy', yx', yy'$). At the level of proper equivalence classes of quadratic forms this becomes a group law.
You could say any group law is defined by a ternary relation $ghk = 1$ on the group.
This fits the geometric description and addition of points on an elliptic curve or Bhargava's interpretation of Gauss's composition.