The deep connection between Jordan theory and Lie theory became apparent in the 1960s through the independent work of Tits, Kantor, and Koecher. After a brief review of this correspondence in the setting of JB*-triples and their associated TKK Lie algebras, we focus on two central notions in JB*-triple theory: tripotents and isometries. We discuss how these notions interact with the Lie structure and how this interplay leads to new structural results and insights into the geometry of JB*-triples.