Non-associative Solomon's descent algebras

Motivated by the observation that universal enveloping algebras of relatively free Sabinin algebras admit natural associative internal products, we revisit the classical relationship between Solomon's descent algebra, noncommutative symmetric functions, and Patras' descent algebras of graded bialgebras from a non-associative perspective. Using the language of algebras over cocommutative connected Hopf operads as a unifying framework, we describe a setting in which these connections extend beyond the associative case. This viewpoint allows us to interpret the free non-associative algebra as a non-associative analogue of Solomon's descent algebra inside a suitably defined non-associative Malvenuto-Reutenauer-type Hopf algebra. We further provide a detailed classification of this object as an algebra, as a coalgebra, and as an associative algebra with respect to its internal product.