The lumbar vertebrae bear synapophyses for articulation with ribs. The pygals bear only transverse processes with distally converging margins and no rib articulation and also no articulation for chevrons.

I have never seen those in mososaurs, but I guess that true pygal vertebrae in all reptiles should have forked distal ending (bilobed) of the transverse processes or at least a groove, a cleft, or a slot at the far end or on the fore-, or hind, or ventral sides of the processes - these all are results of overgrowing or contacting to paired pelvic lymph hearts, which have thick muscular walls. Otherwise the transverse processes are usually smooth and roundish in shape, especially distally, as if they grow without meeting any obstacle.

No... mosasaurs do not have that morphology... simply converging margins terminating in a somewhat blunted end.

interesting! either they had no lymph hearts, or the transverse processes are short in relation to the diameter of the body? well, I do no know... would be interesting to have a look

No lymph hearts IMO. The transverse processes are long at the base of the tail, which was apparently heavily muscled. http://www.bioone.org/doi/abs/10.1666/09023.1?journalCode=pbio

another one    http://www.jstor.org/discover/10.2307/1441880?uid=3739920&uid=2129&uid=2&uid=70&uid=4&uid=3739256&sid=21104875142103

Hello everyone, to start with the question - pygal vertebrae are a specialized form of caudal vertebrae that lack an articulating/or fused haemal arch or chevron bone, but possess a transverse process that does not have an articulating rib.  Typical caudal vertebrae have chevron bones (fused or articulating depending on the kind of mosasaur), and transverse processes without articulating ribs.  Presacral or dorsal vertebrae (the term lumbar is really a mammalian term reflecting specializations of the axial skeleton in derived synapsids) have shorter transverse processes with articulating ribs, and no chevron bones or haemapophyses.  The pygal region in mosasaurs is unique in that there are usually a rather large number of such vertebrae in the caudal series, potentially due to the loss of a sacroiliac joint - the 2 sacrals must go somewhere and so it is possible that in mosasaurs they have become part of the "pygal series" thus upping the count.  In more primitive sistergroups to mosasaurs there is a good sacrum (2 vertebrae) and 2 pygals prior to the typical caudal anatomy; this is why in derived mosasaurs the pygals are considered to be a specialized form of caudal vertebra.  Many modern lizards have 0 (Heloderma) to 1 (Varanus) or 3-4 (Iguanians, etc.) vertebrae at the base of the tail that display a pygal-like anatomy.  However, those same vertebrae in modern lizards usually possess grooves or forks that support lymph node networks - these are not apparent in mosasaur pygals.  It a reasonable supposition that the loss of the sacroiliac joint modified numerous soft tissues systems including the lymph tissues and their placement on the tips of the transverse processes of anterior caudals such as the pygal series.  Interestingly, the reverse is true for snakes, where despite the loss of the sacroiliac joint, the postcloacal vertebrae in proximity to the pelvic elements still possess lymphapophyses. Just to keep it confusing, snakes also present small hemapophyses on the first lympapophysis bearing postcloacal vertebra.

Thank you very much to all of you for your answers!

Yes, actually, correlation between the loss of sacrum, increasing size of the lymph hearts and the number of pygal vertebrae bearing traces of lymphatics on their transverse processes in Squamata is perfect.

I'd agree with everything Polcyn and Caldwell said (short answer: mosasaurs don't have true lumbar vertebrae, but retain distal rib facets on all precloacal transverse processes), except not sure about generality of MC's "snakes also present small hemapophyses on the first lympapophysis bearing postcloacal vertebra". There's at least one good pygal in Wonambi, with no haemapophyses at all and only a weak groove near the tip of the pleurapophysis.