Jordan pair

Definition

Let $F$ be a field of characteristic different from $2$, and let $V^+$ and $V^-$ be vector spaces over $F$. For a sign $\sigma \in \set{+, -}$, write $-\sigma$ for the other sign. A Jordan pair structure on $(V^+, V^-)$ is a pair of trilinear maps $V^\sigma \times V^{-\sigma} \times V^\sigma \to V^\sigma$, for $\sigma = \pm$, both written $(x, y, z) \mapsto \set{x, y, z}$. The conditions are, for each $\sigma$ and all $u, x, z \in V^\sigma$ and $v, y \in V^{-\sigma}$:

1. $\set{x, y, z} = \set{z, y, x}$;
2. $\set{u, v, \set{x, y, z}} = \set{\set{u, v, x}, y, z} - \set{x, \set{v, u, y}, z} + \set{x, y, \set{u, v, z}}$.

In clause 2, the inner product $\set{v, u, y}$ is the one on $V^{-\sigma}$.