Leonard triples from Leonard pairs constructed from the standard basis of the Lie algebra (Q445862)

A Leonard triple is a triple of (semisimple) complex matrices, any given two irreducible tridiagonal in the eigenbasis of the third, extending the well-known concept of a Leonard pair. Perhaps the best known example of a Leonard pair is the pair \((\pi(h), \pi(e)+\pi(f))\), where \((h,e,f)\) form the standard basis of \(\mathfrak{sl}_2(\mathbb{C})\) and \(\pi\) is any nontrivial irreducible representation of it. The authors determine, via some lengthy computations, the Leonard triples extending this Leonard pair.