The Cadillac CTS is one of those cars that's sort of succeeded in spite of its own parent company. While most alphabet-soup acronyms these days are just a word-mash of numbers and letters, the CTS's moniker reveals an intentionally little-known secret of its anscestry. CTS stands for "Catera Touring Sedan," referring to the sadly rebadged and badly marketed former Opel that preceded it. The Opel Omega-derived Catera was a theoretically good idea -- a Euro-type BMW competitor with an all-American badge -- let down by the fact that the Opel Omega was inherently awful to begin with. However, a name change and a new platform did wonders for the "Catera," making it the CTS the legit BMW competitor that it was always meant to be.
There's no way GM could call the CTS a BMW competitor without making it rear-wheel-drive -- several manufacturers, including Cadillac, tried taking on the Germans with front-wheel drive, but nobody who knew anything about BMWs could take a front-driver seriously. Ironically enough, in order to create a BMW competitor, GM used first a TH4L30-E in the Catera and then a TH5L40-E automatic transmission to send power to the rear wheels. What makes that ironic is that both of these transmissions were also sold to BMW, and used in contemporary 3-Series and 5-Series cars. The same was true of the Getrag five-speed manuals used in base cars. Apart from strange bedfellows, all of these transmissions had something else in common, too.
A vehicle with all four wheels on the ground is always turning the transmission's output shaft, even if it's in neutral. The problem with that it that the gearset in the transmission, even in an automatic, is also connected to the output shaft; so, even in neutral, shafts are spinning in bearings and gears are meshing against each other. That requires lubrication, just as it would if the car were moving under its own power. The CTS's transmissions, like most these days, use oil pumps driven off the drive end, or input shaft. That means that if the car is rolling in neutral, half the transmission is moving, but the oil pump isn't. For that reason, it won't take long before these transmissions grind themselves dry and self-destruct while being towed with all four wheels on the ground.
Without doing any work to the car, you have to keep the rear tires off the ground while towing. That means either loading the car completely onto a car trailer, or backward onto a car dolly with the front wheels on the ground. The backward dolly method is the one suggested by Cadillac in the manual, and will work well here once you get the procedure down. The design of the rear subframe, which holds the differential and rear suspension, even offers a couple of fairly convenient points to connect the dolly safety chain. You shouldn't find it too difficult to reach under the car once it's up on the dolly and loop the safety chain over the rear subframe crossmember, behind the differential. So, the backward-loaded dolly method will likely be the method of choice for most.
Aside from dolly towing, you have a few other options. If you need to flat tow the car with all four wheels on the ground, you can disconnect the rear wheels by unbolting the front and rear sections of the driveshaft at the rubber isolator in the middle. But that's a pain, and isn't free of risk of damage, so this is only really appropriate for emergencies. You can also purchase and install an auxiliary electric transmission oil pump to keep oil flowing while flat towing; but these pumps are expensive, and after the cost of fitting a tow bar to the front of the car, will wind up costing you more than a decent car dolly. The last option is loading the car all the way onto a car trailer. This might be a good option if you do a lot of RV driving, and use the car somewhat infrequently, since it will reduce unnecessary wear on the car and its tires. However, it's much easier and quicker to load and unload the car from a dolly, and you don't need an extra 40 feet of space behind the trailer to do it.
