Let me give you a counter-example with an associative $\beta$, i.e. a Frechet algebra for which the exponentials do not exist in general: The main reason is that a Frechet algebra needs not to be locally *multiplicatively* convex. On the Weyl algebra with two generators $Q$ and $P$ subject to the commutation relations $[Q, P] = 1$ there are several locally convex topologies possible such that the completion yields a Frechet algebra. Now it is well-known that none of them can be locally multiplicatively convex. In fact, one can show that e.g. the exponentials of quadratic expressions in the generators like $Q^2$ do not converge.

If of course you have a Frechet algebra which is locally multiplicatively convex then you have an entire calculus and hence in particular exponentials of all elements.