Updating Local MSSQL server from Linked MSSQL server

Trying to update ao  Local Server Database from a Linked Database and keep getting multi-part identifier errors.
MSSQL Servers
Local Server: SQL3
Linked: ILL-SSQL

USE sagedata
Update [SQL3].sagedata.dbo.transactions
Set [SQL3].sagedata.dbo.transactions.illnumber = [ILL-SQL].sagedata.dbo.transactions.illnumber
FROM [ILL-SQL].sagedata.dbo.transactions
INNER JOIN ILL-SQL ON [SQL3].sagedata.dbo.transactions.transactionnumber = [ILL-SQL].sagedata.dbo.transactions.transactionnumber
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