iRule assistance with redirecting to a different pool and URI

Hello you have a conflicting statement in your question :)

You cant have http://oldserver.contoso.com/department/oldsitename/ redirect to http://newserver.contoso.com/business/department/newsitename/ as that would mean the client is following that redirect and the browser would show: http://newserver.contoso.com/business/department/newsitename/

So I think you need to decide if you want the client to still be accessing the old hostname but with a new url or if you want a whole new host and url.

Once I know that we can build the irule. In short a redirect redirects the client browser, so I dont think you want a redirect if you want the old host name.