I think they do it outside on the grounds somewhere, not in the Dome of the Rock. It’s regarded as a sort of terror activity and not encouraged. But I did read it had happened.
During our RCIA one of the teachers told us about how it would have looked at the Temple during times of large sacrifices. The priest would have been dousing the altar in blood which was then washed and drained out the side. So there would have been blood and water pouring out the side of the temple.
The Second Temple had a large square bronze sacrificial altar in front of it. It was huge- something seventy feet on a side, and surrounded on each sides with places for people to stand. Hundreds could probably squeeze around the altar for a single sacrifice on the high Holy days.
So the Altar for sacrifice was completely separate from the Temple building, which apart from the inner sanctuary had administrative offices for the priests and probably sleeping quarters.
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