Involved in the biosynthesis of the spirotetramate antibiotics pyrroindomycins. Catalyzes the intramolecular cyclization forming the dialkyldecalin moiety in pyrroindomycins, via an endo-selective [4+2] cycloaddition reaction. 4-[(2E,7S,8E,10E,13R,14R,16E,18E)-14-ethyl-7,13-dihydroxy-2,16,18-trimethylicosa-2,8,10,16,18-pentaenoyl]-2-methylidene-5-oxo-2,5-dihydro-1H-pyrrol-3-olate = 4-[(1R,2R,4aS,5S,8aR)-2-[(2R,3R,5E,7E)-3-ethyl-2-hydroxy-5,7-dimethylnona-5,7-dien-1-yl]-5-hydroxy-1-methyl-1,2,4a,5,6,7,8,8a-octahydronaphthalene-1-carbonyl]-2-methylidene-5-oxo-2,5-dihydro-1H-pyrrol-3-olate Binds 1 FAD per subunit. The FAD cofactor does not cycle between oxidized and reduced forms and therefore does not have a typical redox role in the reaction; it plays a structural role essential for PyrE3 catalysis, to maintain the requisite geometry of the protein and position the 1,3-diene and dienophile groups through specific interactions with the substrate. kcat is 223.2 min(-1). Antibiotic biosynthesis. Homodimer. Cells lacking this gene lose the ability to produce pyrroindomycins. Belongs to the PheA/TfdB FAD monooxygenase family. Truncated N-terminus.