Catalyzes the ATP-dependent formation of a covalent bond between the amino group of glutamate and the gamma-carboxyl group of the C-terminal glutamate residue in LysW. [amino-group carrier protein]-C-terminal-L-glutamate + ATP + L-glutamate = [amino-group carrier protein]-C-terminal-gamma-(L-glutamyl)-L-glutamate + ADP + H(+) + phosphate Binds 2 magnesium ions per subunit. Amino-acid biosynthesis; L-arginine biosynthesis. Homotetramer. Interacts with LysW. Belongs to the RimK family. LysX subfamily.