Product theorems for alternative algebras and some of their applications

Product theorems are available for associative algebras over a scalar ring containing $\frac{1}{6}$ as well as for such algebras of rank $3$. We prove the exact analog of a product theorem for alternative algebras of rank $3$ and describe the identities in three variables for the alternative algebras Lie nilpotent of a given degree. We also prove an analog of a product theorem in a general form without any restriction on the algebra rank and show that there is no exact analog of a product theorem for alternative algebras. The connection between the concepts of the Lie and strong Lie nilpotency is studied for a given algebra of rank $3$ and its multiplication algebra.