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We find the index of symmetry for all solvable three-dimensional Lie groupswith a left-invariant metric. When combined with the work of Reggiani onunimodular three-dimensional Lie groups, the index of symmetry is then knownfor all three-dimensional Lie groups with a left-invariant metric. Similar tothe unimodular case, for every solvable three-dimensional Lie algebra there isat least one left-invariant metric for which the index of symmetry is positive.Also, the index is never equal to 2.

The index of symmetry for a left-invariant metric on a solvable three-dimensional Lie group