Catalyzes the ATP-dependent formation of a covalent bond between the amino group of alpha-aminoadipate (AAA) and the gamma-carboxyl group of the C-terminal glutamate residue in LysW. [amino-group carrier protein]-C-terminal-L-glutamate + ATP + L-2-aminoadipate = [amino-group carrier protein]-C-terminal-N-(1,4-dicarboxybutan-1-yl)-L-glutamine + ADP + H(+) + phosphate Binds 2 magnesium ions per subunit. Amino-acid biosynthesis; L-lysine biosynthesis via AAA pathway; L-lysine from L-alpha-aminoadipate (Thermus route): step 1/5. Homodimer. Belongs to the RimK family. LysX subfamily.