Cramer's decomposition theorem states that if $X$ and $Y$ are independent real random variables and $X+Y$ has normal distribution, then both $X$ and $Y$ are normally distributed. I've seen a few proofs of this result which involve grunt work in the realm of complex analysis. I was wondering if there is an intuitive proof of this result. I would like to see a proof that exploits the unique nature of the normal distribution. For example, can one derive the result from the fact that on the real line, the normal distribution maximizes entropy for a given mean and standard deviation?

Edit: The complex analysis proof I'm thinking of uses the fact that if $E[\exp(\alpha X^2)]<\infty$ for some $\alpha>0$ and the analytic continuation of the characteristic function of $X$ is nonzero, then $X$ is normally distributed.