Producing high purity lithium solution from a lithium source containing dissolved Na+, Ca2+, and Mg2+, by: passing the source into and out of a bed of sorbent composed of hydrated alumina intercalated with LiX, preferably LiCl, to extract lithium from the lithium source into the sorbent; washing the bed of sorbent with dilute aqueous LiCl to remove lithium from the sorbent to obtain a lithium eluent of increased Li+ concentration; subjecting this eluent to nanofiltration to produce a lithium permeate from which Ca2+, Mg2+, and other nanofilterable components are concurrently removed, yielding a permeate solution with 25% or less, and a retentate solution with 75% or more Ca2+ and Mg2+, as compared to the eluent from washing; and subjecting the permeate solution to a particular forward osmosis yielding a solution having 13,000-25,000 ppm dissolved lithium. Specified optional steps and new features can be used to increase lithium concentrations and purity.