Multiple family members can definitely be part of the same pool, but that's not what the OP was asking about. Obviously pools should have a written contract, and it's probably even more important if there are multiple members of the same family in the pool and one of them is running the pool.
Even if you don't have a written agreement the IRS is probably happy enough to believe that if you share a lottery prize with coworkers it's because you had a mutual agreement requiring you to share it, rather than simple generosity. When you share with family they seem to start from the assumption that you're simply being generous or acting out of a sense of familial responsibility rather than following through on a pre-existing agreement that requires you to share. The former is a contractual obligation, whereas the latter is a gift that's subject to gift tax rules. Since other pool members would have reduced shares if some of the shares go to multiple family members a written agreement might not be necessary, but leaving it to chance might cost millions in gift taxes.
There's also no reason that you couldn't share your share of a pool with family members. You just have to be able to document the agreement that existed before the ticket becomes a winner. The IRS usually looks for a quid pro quo agreement, where what you've promised to family members is based on a similar promise they've made to you. You can't just promise them a portion of something that you might win, because that's just a gift that you'll make if you do win. What you can do is gift part of your share to others while your share is still only worth part of the few dollars you paid to the pool, since that's well below the annual gift tax exemption. If you do that you should expect that you'll have to prove that you really made an irrevocable gift that would apply to a winning ticket that's only worth 10k or 250k, as well as a jackpot winner that's worth millions. I'm guessing that most of the people who really would share part of a multi-million dollar prize would want to keep all of a prize that's not enough to spend the rest of their life on easy street, and I'm sure the IRS thinks so, too. As a result, I figure that any gift of a share has to be accompanied by an agreement that makes it easy for those receiving that gift to know if you've won and to enforce the agreement.