2004 Z4 2.5i, manual trans & differential

[BMWrdr]

I recently purchased a 2004 2.5i with a 5 spd manual transmission. It has had multiple owners over its 18 year life, and has 58 K miles. The shifting is rather stiff and I believe it’s time for a change of oil in the gearbox and the differential. Finding out what type/kind of oil and the amount required for each application is at best difficult as the manual is silent on the subject and the information on the web and on this form is scattered and not very specific. I want to use synthetic oil. I have raced cars in the past and became a total fan of synthetic oil. I’m a long time rider of BMW motorcycles and I use synthetic oils in my bikes. So lend some helpful information to this old guy who is a newbie here. I’m thinking of using Red Line D4 ATF in the trans, or perhaps Royal Purple. Your recommendations AND the amount I’ll need for the transmission would be appreciated. For the differential I’ll await your input on type and amount please.