Santiago and James

Easter has passed. Perhaps it is relevant to inquire some rather
religious questions.
In saint names we find English Joseph, John, Peter, Paul, James
becoming the Spanish Jose, Juan, Pedro, Pablo, Jaime.
All obviously plausible.
But my question centers on James which other than Jaime has Santiago
as equivalent. I saw one claim from which it stated Santiago comes
from Latin Sanctus Jacobus. Which would equate Jacob with James.
But is this attested? Or there may be possible semantics within or
outside IE which would give us the equation of Spanish Santiago with
English James?
Thanks for any interesting answer (from Miguel or anyone).

Loreto