This talk is intended to explore the linear extendibility of surjective isometries between the unitary sets of two unital JB$^*$-algebras. The quoted question will illustrate the role played by the unitary elements in the algebraic structure of a unital JB$^*$-algebra. In this framework, we shall draw special attention to the connected components of the unitary set, and complete our study by providing an accurate description of the surjective isometries between the principal components of two unital JB$^*$-algebras.