How many miles in your car?

If > 200K, replace the hydraulic tensioner.

Anyway, a little bit of play is OK, just lubricate the pivot points with some engine oil. The pulley: you can buy just the pulley alone. I use ONLY INA brand.

Info in bimmerfest E39 forum.

PS: Consider cooling overhaul if > 150K miles.

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1998 E39 528i 5sp MT
2006 E53 X5 3.0 6sp MT