In the free special Jordan algebra $SJ[X]$ is valid the equality
$$T(x,y,z,t) = \frac{1}{4}([x,z] \circ [t,y] + [x,t] \circ [z,y]),$$
where $T(x,y,z,t) = (xy,z,t) - x(y,z,t) - y(x,z,t)$. Here $[x,y] = xy - yx$, $(x,y,z) = (xy)z - x(yz)$ (associator) and $\circ$ is the Jordan product in the algebra $F(X)^+$, ie, $x \circ y = \frac{xy+yx}{2}$.
Comments: I'm now starting to study Jordan algebras and I am having trouble understanding how to do operations on such algebras. Several authors say that the above equality is easy to verify, but replacing the product $\circ$ I am not able to verify. If you can give me some tips on how to do this, thank you very much.