The Galois algebra with Galois group which is the automorphism group

Let B be a Galois algebra over a commutative ring R with Galois group G. Then it is shown that G=AutR(B) if and only if B is commutative with no idempotents but 0 and 1, or B≅R⊕R where R contains no idempotents but 0 and 1.