In House of the Dragon, season 2, episode 4, shortly before the final battle between three dragons starts, King Aegon arrives with his dragon Sunfyre.
The first two commands King Aegon gives to Sunfyre are spoken in the Common Tongue (English):
Is it not an established fact that dragons are commanded by being spoken to in High Valyrian?
It's also worth noting that earlier in the same episode we see that Aegon's Valyrian is pretty lousy. During a council meeting Aegon finds out that Aemond and Cole have been making military plans behind his back. Aemond starts speaking to Aegon in notably smooth Valyrian. Aegon's response in Valyrian is given in the subtitles as:
I can have to make a war?
He clearly struggles with getting even this one sentence of nonsensical Valyrian out.
Granted, he probably is capable of using basic common phrases for commanding his dragon in Valyrian, but given he's so uncomfortable with it it's only natural for him to default to the common tongue, especially in the middle of a battle with no one else around to hear him but the dragon.
In this episode and others we see that Aegon isn't the wisest of kings. The mere fact that he impulsively jumps onto his dragon to join the fight after being made to feel useless by his council and mother is evidence enough of that.
We also can't actually say definitively whether Sunfyre understood him or even listened to him. All he said was to go forward and faster. The dragon roars when he says forward, but it's already going forward, and sure it looks like it might go a bit faster after, but it certainly isn't obvious that it's because of the verbal command. I'm not arguing that Sunfyre didn't listen to him, just that we didn't see anything definitive in the episode that proves it. If he said "breathe fire" and then Sunfyre did it, then sure, that would be proof.
Plus animals that are ridden, such as horses, are usually controlled via non-verbal commands, such as via movements of the reins or of the feet. Horse riders will very often throw in verbal commands, but non-verbal cues are the main way they control their horses. We see Aegon and the other dragonriders engaging with their dragons in a similar fashion, such as by pushing forward on their handler or leaning to either side, so it's reasonable to assume that at least some component of their communication is non-verbal. They aren't constantly shouting which direction the dragon should turn after all.
However we've seen in the show and the books though that dragons are somewhat intelligent. In Game of Thrones (S06E02) Tyrion Lannister said:
Dragons are intelligent, more intelligent than men according to some maesters.
So there's no reason to believe that they can only understand Valyrian.
No, it's not strictly necessary to command dragons in High Valyrian. While it's a common practice, dragons can understand and respond to commands in the Common Tongue as well.
In the book A Storm of Swords (Daenerys I), we see an example of this flexibility:
"Stop that, Rhaegal," Dany said in annoyance, giving his head a swat. "You had the last one. I'll have no greedy dragons."
Here, Daenerys commands her dragon Rhaegal in the Common Tongue. However, she does use High Valyrian for specific commands, as explained in this passage (emphasis mine):
"Dracarys?"
All three dragons turned their heads at the sound of that word, and Viserion let loose with a blast of pale gold flame that made Ser Jorah take a hasty step backward. Dany giggled. "Be careful with that word, ser, or they're like to singe your beard off. It means 'dragonfire' in High Valyrian. I wanted to choose a command that no one was like to utter by chance."
Daenerys deliberately chose to train her dragons to respond to certain Valyrian commands to prevent accidental triggering. This precaution ensures that a dragon won't inadvertently harm someone if they happen to say a common word like "fire" in the Common Tongue.
It's worth noting that commanding dragons isn't as simple as speaking to them in any language. George R.R. Martin has explained that controlling dragons is a complex and mysterious process, more akin to taming wild animals than giving simple verbal commands. He compares it to horse training, saying:
It's not a simple process, it's a dangerous process […] We all know the bronco buster kind of thing that we see in western movies […]
This suggests that while speaking High Valyrian might be part of dragon communication, it's not the only factor in controlling them. It's likely a complex combination of ability (inherited or otherwise), various communication methods, and likely magic—not just language.
So, while High Valyrian is often used for dragon commands, it's not an absolute requirement. Dragons can understand and obey commands in other languages, including the Common Tongue, as demonstrated by King Aegon II's interactions with Sunfyre in House of the Dragon.
