segregated witness – Do non-Segwit nodes reject Segwit transactions with invalid signature?

If a miner included a Segwit transaction to a block with invalid signature, it’d be rejected by Segwit nodes, but not by non-Segwit nodes.

Indeed. It would have to reach the miner so either sent to them directly or propagated to the miner by non-SegWit nodes. SegWit nodes would reject it and not propagate it.

Would the latter consider the transaction valid? If yes, wouldn’t this result in a chain split?

Full nodes enforcing SegWit rules would reject a block with an invalid SegWit transaction in it as would miners enforcing SegWit rules. What would happen in your scenario is the miner would think they’ve successfully mined a block but full nodes enforcing SegWit rules would reject it and miners enforcing SegWit rules would refuse to mine blocks on top of it. So it can cause temporary confusion amongst full nodes not enforcing SegWit rules but the longest chain with most proof of work would end up not including that block and that is the chain that even the full nodes not enforcing SegWit rules would end up following.

Definitions aren’t always clear (e.g. re-org, chain split) but there wouldn’t be a lasting chain split unless a subset of miners refused to reject this block that didn’t meet SegWit rules and continued to mine in perpetuity blocks on top of it. This would be a very costly endeavor as they would effectively be mining an altcoin and not what is considered Bitcoin.

If a miner included a Segwit transaction to a block with invalid signature, it’d be rejected by Segwit nodes, but not by non-Segwit nodes.

Indeed. It would have to reach the miner so either sent to them directly or propagated to the miner by non-SegWit nodes. SegWit nodes would reject it and not propagate it.

Would the latter consider the transaction valid? If yes, wouldn’t this result in a chain split?

Full nodes enforcing SegWit rules would reject a block with an invalid SegWit transaction in it as would miners enforcing SegWit rules. What would happen in your scenario is the miner would think they’ve successfully mined a block but full nodes enforcing SegWit rules would reject it and miners enforcing SegWit rules would refuse to mine blocks on top of it. So it can cause temporary confusion amongst full nodes not enforcing SegWit rules but the longest chain with most proof of work would end up not including that block and that is the chain that even the full nodes not enforcing SegWit rules would end up following.

Definitions aren’t always clear (e.g. re-org, chain split) but there wouldn’t be a lasting chain split unless a subset of miners refused to reject this block that didn’t meet SegWit rules and continued to mine in perpetuity blocks on top of it. This would be a very costly endeavor as they would effectively be mining an altcoin and not what is considered Bitcoin.