I don’t like the turn check. A king is a great card for you to bet at – he’s 3-bet and called a 4-bet, he very easily has a king in his range here. So if he has a king you’re getting value, and most likely still ahead. If he has backdoor diamonds or (as it happens) a straight draw, you want to charge him for that draw.
Pretty lose pre-flop by him, he will never fold the flop, but could definitely fold on the turn.
This really depends on what you know about the villain in my mind. He 3-bet pre, then called a 4-bet with 45 suited. That’s really loose, and if you know a player is really loose you need to put maximum pressure. Sure, he could have trips or a set, but he’s also got overpairs, draws, and could have also have ran into a K. There’s only a few hands that beat you and a lot that you beat.
Are you ever folding AA here? Probably not. I probably jam turn and put max pressure on him. You gotta punish the guys playing this loose when you have a strong hand.
I think the comments above are fair. The pot has nice value and I think you should be happy to take it down if he doesn’t come along. Did you check the turn to induce river action or were you afraid of the 6?… just unfortunate that he hit his 34% draw. Not knowing the opponent, its hard to say that you didn’t have him right where you want him, but I agree with the other guys above… its a nice pot… attack weakness. Not sure how you can put him on 4-5 there. Was it in his range to have a 6? Based on pre-flop action, would have been tough to put him on that. Just a hand that got lucky. It’ll happen.