I use 10 w 40 Power 1 4t racing oil on my America and it has worked flawlessly in all kinds of temperatures. Living in Wisconsin that's what we get.
I have a 2023 Bonneville T120 Black and live in NE Indiana; local Triumph dealer recommended Castrol 10 w 50 Power 1; bought 4 quarts of that and will use for my first oil change at 600 miles along with a Triumph oil filter.
It really just depends on the temps where you live and ride. If I lived in the desert where air temps were consistently over 30 C, then I would use a 10-50 Weight oil. Just follow the manufacturers recommendation and you'll be fine. Brand I'll leave up to your.
Never known Modern Triumphs to use Castrol at present they use silkolene fully synth. But again like mentioned it all depend on climate and atmospheric climates. I would ask around some reputable dealers just to get a clue to what is recommended. Joe
I actually just bought a bunch of Shell Advance 4T Ultra 10-40 motorcycle oil for a very good price. If its good enough for Ducati, it should be fine in the Triumph. It's a full sync oil, SN, Jaso MA2 spec. The local dealer uses Liguid Moly