This question contains many spoilers for Tenet.
I'm trying to work out exactly what happens in the car chase sequence. I think I understand where most of the characters go, but there's a couple of objects where I don't understand what happens to them.
The main one is the silver box that later turns out to be part of the algorithm. It seems that this is what happens:
Initially the protagonist has the algorithm with him in his car. Then at some point the silver car un-flips and joins the chase.
In a third car, Sator appears to be counting down on his fingers (though from Sator's point of view he's counting up), and the protagonist throws him the orange case that originally contained the algorithm. However, we later find out that he threw the empty orange case to Sator while almost simultaneously throwing the algorithm into the silver car.
Later, the protagonist goes through the turnstile, exits the building inverted, and finds the silver car.* For some reason, he doesn't seem to realise that at this point the algorithm is (as far as I know) somewhere in this car.† ‡ He then drives off and enters the car chase. The algorithm then jumps up from somewhere and flies into the BMW, which is how we know he threw it in there in forward time.
He then crashes the car, and Sator catches up with him and sets it on fire. But as the diagram above shows, the algorithm isn't in the car at this point - in the inverted time line, it's already jumped into the BMW.
After that the protagonist loses consciousness, and when he wakes up he says that Sator has the algorithm. Later we find out that this is true, since all the algorithm's components have been assembled at the final battle.
So in forward time, the algorithm seems to end up sitting in the silver car, parked outside the turnstile facility, which has been taken over by Ives' team. The question is how Sator's people get hold of it after that. It happens off screen, but is it explained or hinted at?
Sator himself can't have picked it up, because he inverts before the protagonist gets into the car. Unless he uninverted later, before inverting again and going back to Vietnam.
The other thing I don't understand is what happens to the orange case that the algorithm was in originally. The (non-inverted) protagonist throws it to Sator. But this Sator is inverted, which means that from his point of view it jumps out of his hands and into the protagonist's. (After that he starts counting upwards on his fingers for some reason.) But how did Sator get hold of it before that, from his perspective? At some point it's by the side of the road, but if it was thrown from the inverted car he wouldn't have had it at that point to pass it to the protagonist. I probably missed something here - which car was it thrown out of, and when? (On re-watching, in reverse-time we see the orange case jump off the curb into the non-inverted Mercedes, which inverted Sator is riding at that point in time. I guess we have to assume that it was thrown out of the car by its non-inverted driver.)
In my headcanon this is an inverted getaway car that Sator's gang left outside the inverted half of their hideout. This explains why the car itself seems to be inverted, and also explains why there's a fancy fast car so conveniently placed at that location. Nope, that doesn't make sense - they would have to invert the car in the future in order for it to be there at that time, but Ives' team owns that facility now. Sator's team could re-capture it in the future and invert the car then, or invert it somewhere else, but then why would they go out of their way to put their inverted car there, knowing that it would be used by the protagonist in the car chase? Unless that's how they got hold of the algorithm, in the future, when they inverted it. But it seems more likely that it's just a normal car and it's just cinematic license that it behaves inverted when the protagonist drives it. (This point continued to bug me, so I've asked another question here.)
†If he did realise that and tried to take it out of the car it probably wouldn't work, because it would make it impossible for him to have thrown it there in the first place. But I guess there's nothing to stop him telling one of his friends to go back through the turnstile and pick it up in forward time. He could then go and enter the car chase, to make sure the algorithm would be there in the first place. I suspect the idea is that he hasn't quite understood how time inversion works, so he thinks the algorithm will come into the Saab when his other self throws it.
‡Having watched it again on the small screen, he does check behind the driver seat when he first gets into the car, so maybe he knows the algorithm should be there after all, and is looking for it. He doesn't seem to look very hard, though, and fairly quickly gives up and drives off.