Local-global divisibility on some commutative algebraic groups

Given a commutative algebraic group A defined over a number field K, one can ask if the local-global principle applies to solutions of equations of the form [n]D=P, with P in A(K). Although no complete answer is known, a cohomological investigation of this problem provides some conditions for a positive, or negative, answer.
A more complete description can be given for algebraic tori and elliptic curves.