A long angry email invites a long careful reply. That usually keeps the heat up, even when every sentence is polite.
A shorter reply that names the problem and the next concrete step cools things faster: what went wrong, what you're doing, when they'll hear again.
Keep the tone flat. No sarcasm, no "as I already said", no matching their volume. One calm paragraph beats three defensive ones.
If you need facts you don't have yet, say what you'll check and by when. Then actually come back on that clock.
There's more on this, including when to take the thread offline, in a calm way to cool down a heated email thread.
Grais · m@grais.ai