Is involved in the reduction of 2,3-digeranylgeranylglycerophospholipids (unsaturated archaeols) into 2,3-diphytanylglycerophospholipids (saturated archaeols) in the biosynthesis of archaeal membrane lipids. Catalyzes the formation of archaetidic acid (2,3-di-O-phytanyl-sn-glyceryl phosphate) from 2,3-di-O-geranylgeranylglyceryl phosphate (DGGGP) via the hydrogenation of each double bond of the isoprenoid chains. a 2,3-bis-O-phytanyl-sn-glycerol 1-phospholipid + 8 oxidized 2[4Fe-4S]-[ferredoxin] = a 2,3-bis-O-(geranylgeranyl)-sn-glycerol 1-phospholipid + 16 H(+) + 8 reduced 2[4Fe-4S]-[ferredoxin] 2,3-bis-O-(phytanyl)-sn-glycerol 1-phosphate + 8 oxidized 2[4Fe-4S]-[ferredoxin] = 2,3-bis-O-(geranylgeranyl)-sn-glycerol 1-phosphate + 16 H(+) + 8 reduced 2[4Fe-4S]-[ferredoxin] Binds 1 FAD per subunit. Membrane lipid metabolism; glycerophospholipid metabolism. Reduction reaction proceeds via syn addition of hydrogen for double bonds. Belongs to the geranylgeranyl reductase family. DGGGPL reductase subfamily. Extended N-terminus.