  This work addresses the questions: (i) Among all left-invariant Riemannian
metrics on a given Lie group, is there any whose isometry group or isometry
algebra contain that of all others? (ii) Do expanding left-invariant Ricci
solitons exhibit such maximal symmetry? Question (i) is addressed both for
semisimple and for solvable Lie groups. Building on previous work of the
authors on Einstein metrics, a complete answer is given to (ii): expanding
homogeneous Ricci solitons have maximal isometry algebras although not always
maximal isometry groups. As a consequence of the tools developed to address
these questions, partial results of Boehm, Lafuente, and Lauret are extended to
show that left-invariant Ricci solitons on solvable Lie groups are unique up to
scaling and isometry.
