A determinantal proof of the Craig-Sakamoto theorem

The Craig-Sakamoto theorem states that if A and B are symmetric matrices, then (a) I − αA − βB = I − αAI − βB for all α, β if and only if (b) AB = 0. There are a number of proofs of this result, the most common based on expansions of the logarithm of (a). The present proof is elementary in that it depends only on determinantal conditions.