Protein-lysine methyltransferase that selectively mono-, di- and trimethylates 'Lys-165' of the translation elongation factors EEF1A1 and EEF1A2 in an aminoacyl-tRNA and GTP-dependent manner. EEF1A1 methylation by EEF1AKMT3 is dynamic as well as inducible by stress conditions, such as ER-stress, and plays a regulatory role on mRNA translation. L-lysyl-[protein] + 3 S-adenosyl-L-methionine = 3 H(+) + N(6),N(6),N(6)-trimethyl-L-lysyl-[protein] + 3 S-adenosyl-L-homocysteine L-lysyl-[protein] + S-adenosyl-L-methionine = H(+) + N(6)-methyl-L-lysyl-[protein] + S-adenosyl-L-homocysteine N(6)-methyl-L-lysyl-[protein] + S-adenosyl-L-methionine = H(+) + N(6),N(6)-dimethyl-L-lysyl-[protein] + S-adenosyl-L-homocysteine N(6),N(6)-dimethyl-L-lysyl-[protein] + S-adenosyl-L-methionine = H(+) + N(6),N(6),N(6)-trimethyl-L-lysyl-[protein] + S-adenosyl-L-homocysteine Interacts with members of the heat shock protein 70 and 90 families and of the TCP-1 chaperonin family, as well as with HSPD1, STIP1 and tubulin; at least some of these proteins may be methylation substrates. Belongs to the methyltransferase superfamily. METTL21 family.