How iterative reflections of monads are constructed

Every ideal monad MM on the category of sets is known to have a reflection Mˆ in the category of all iterative monads of Elgot. Here we describe the iterative reflection Mˆ as the monad of free iterative Eilenberg–Moore algebras for MM. This yields numerous concrete examples: if MM is the free-semigroup monad, then Mˆ is obtained by adding a single absorbing element; if MM is the monad of finite trees then Mˆ is the monad of rational trees, etc.