Non-chlorinated brake cleaner, seriously. I use the stuff they sell at NAPA and buy it by the case for cleaning derailleurs, etc.. My friend owned a garage and even though his mechanics used those paper mats grease from the mechanic's shoes would inevitably get on the carpet along the door sill or worse yet grease from their clothing or arms would get on the upholstery. They use brake cleaner to get it out and it does it RIGHT NOW. They turned me on to it about 10 years ago and I've used it ever since. It even works on road tar but it takes several applications and still leaves a slight discoloration but regular grease stains come right out. Test it on an area that's not visible like under the edge of a seat or where the carpet goes up under the dash to be safe but I haven't had a problem yet. I keep a can in my old work car, I use it to pick up bikes I buy and even though I put them in a contractor bag before loading them I always seem to get a smudge on the carpet somehow. No problem though.
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".....distasteful and easily triggered."