Standard Envelopes of Commutative Triple Systems

Under certain constraints on the characteristic of a field $\Phi$, the commutative standard enveloping $q$-algebra $B$ of a commutative triple system $A$ over $\Phi$ is defined. It is proved that

• 1) if the algebra $B$ is simple, then the system $A$ is simple;
• 2) if the system $A$ is simple, then $B$ either is simple or decomposes into the direct sum of two isomorphic simple subalgebras (as of ideals).