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"She's beating you up verbal-ninja style. She's ripping you to shreds with razor blades of icy sisterhood."
Uh, no, she's being a catty bitch. The stuff about jealousy and blame is true, though.
"WE" women don't do this to each other. Immature women (and men) do this to each other (anyone remember middle school?) because it gives them a sense of power. It is up to the LW to let the ex have the power or not.
I would say NO to the carpool. The LW doesn't have to prove herself to the ex or anyone else, and I am sure the BF would be happier not to hear two women exchanging tense remarks leading to veiled (or not) insults. As for putting the LW down in public, do those nearby chime in, do they grin, or do they have that "I'm not looking at that train wreck" expression? If the latter, the LW is doing all she needs to do. Otherwise, a well-placed expressionless "Whatever" or a quizzical "Why do you say that?" should put her on the spot...and in a way that makes clear she's in that spot of her own doing. The problem with coming up with bitchy retorts is that the whole situation will spiral downward and they never sound as good as they do in your brain anyway.
The ex doesn't like LW? Nowhere is it written that everyone must like you for you to be a good person. If the thought of the ex's disdain bothers the LW, she should practice following each said thought with the words, said slowly "So. Fucking. What?"