A complete proof can be found in the first few pages of
https://mathscinet.ams.org/mathscinet-getitem?mr=977759
Jensen, Christian U.(DK-CPNH); Yui, Noriko(3-TRNT) Quaternion extensions. Algebraic geometry and commutative algebra, Vol. I, 155–182, Kinokuniya, Tokyo, 1988.
I will scan the relevant pages and post them here as soon as electricity is restored in my office.