I have a '98 528i so likely different for 540i. I started out a little worried that this would be more involved than I expected. Unlike the e34 where there is a hex, our e39 has a torx on the tensioner (T-47). So I had to make a trip to the store to purchase a set. The tricky part is the a/c belt where there isn't enough clearance to get a ratchet and torx on the tensioner. The radiator shroud blocks a straight shot. So I used a torx socket (T-47 size) and then got my small pipe wrench (see pic). It gripped successfully and I could loosen the belt. I couldn't see any other way and I was not about to take apart the radiator shroud. Once the main belt is loosened, it can be wiggled over the fan very easily. No need to remove the fan. Here's a step by step and a picture of my wrench jig to get it to work -

1) Draw picture of serpentine path of belt (very very important or you will regret it during install!!!).

2) Remove front plastic panel using Phillips screwdriver.
3) Pop tensioner pulley caps with screwdriver to reveal the torx

(see pic - the view is from the bottom looking up). The pic is the main pulley tensioner.

4) Tensioner has arrows showing you the direction you pull to move the tensioner and loosen the belt.
5) Remove the accessories belt first.
6) On main belt, pull loosened belt up over the fan.
7) Replace in reverse order.
8) Watch for fingers when releasing the tensioner (or if the wrench slips)

Getting the main belt back on is tricky. It requires some jigging and working. But with some time and patience, you will eventually get it. I had to review the diagram many times during the process. My brain can't comprehend it upside down. If you have help from above or even below, it would be easier.

Our '98 has a little under 38K miles. I was surprised at how cracked the belts were. A definite replacement item for maintenance II.

Send me an email at griffs67@yahoo.com if you have any questions.

Patrick