Lysophosphatidic acid acyltransferase which functions in phosphatidic acid biosynthesis. May regulate neutral lipid accumulation and participate in the regulation of lipid turnover in vegetative cells. May possess additional triacylglycerol lipase and phospholipase A2 activities in vitro (By similarity). a 1-acyl-sn-glycero-3-phosphate + an acyl-CoA = a 1,2-diacyl-sn-glycero-3-phosphate + CoA The HXXXXD motif is essential for acyltransferase activity and may constitute the binding site for the phosphate moiety of the glycerol-3-phosphate. Belongs to the peptidase S33 family. ABHD4/ABHD5 subfamily.