Associative and Jordan Lie nilpotent algebras

We look at the interconnection between Lie nilpotent Jordan algebras and Lie nilpotent associative algebras. It is proved that a special Jordan algebra is Lie nilpotent if and only if its associative enveloping algebra is Lie nilpotent. Also it turns out that a Jordan algebra is Lie nilpotent of index $2n+1$ if and only if its algebra of multiplications is Lie nilpotent of index $2n$. Finally, we prove a product theorem for Jordan algebras.