Do you write North South East West with a capital letter? Why do you write then: He lives in north London?

Generally no,you don't write them with capitals.

The exception is place names like North Daktoa or the South Pole.

See when you use north in the way you said in the second sentance its a common noun, like car or sun.

Its confusing, I know.

Does this help?

They should be capitalized when referring to a part of a country or province, but not when used as a direction.

For example:
I live in Northern Ontario.
Drive west until you see the sign.

yeah, so place names capatilized all else normal text.

So you don't write :I live in the North?

Yep that's exactly right!

Mark, no. You would write --

I live in the north of Germany.
or
I live in the southeast part of the city.

But for specific place names or recognized parts of a country or state, you'd write ==

She is from Southern California.
or
She came from the South. (meaning the states in the recognized southern part of the US)

http://grammar.ccc.commnet.edu/grammar/capitals.htm
See Section 4.

The words "north," "south," "east," and "west" are generally not capitalized unless they are being used as part of a proper noun or as the first word in a sentence. Here's a breakdown of when to capitalize them:

1. As part of a proper noun: When a specific place or region is being referred to, such as "North America," "South Africa," "East Asia," or "West Indies," the directions are capitalized because they are integral to the proper name.

2. At the beginning of a sentence: Like all words in English, when "north," "south," "east," or "west" are the first word of a sentence, they should be capitalized.

On the other hand, in your example sentence, "He lives in north London," the word "north" is not capitalized because it is being used as a general directional description rather than part of a proper noun. It is simply indicating the specific part of London where the person lives, without referring to it as an official place name.