I never study deep into Fallout universe, all I know about Brotherhood of Steel is they are bunch of nice guys in Fallout 3.

In Fallout 3, BoS did not give any bad impression to me. But BoS in Mojave...they are really looks like fanatic as Mr. House suggested.

If they are so obsess toward tech, shouldn't they belong to Outcast?

I think that the oudcasts in fallout3 left the brotherhood there becaurse they didnt like the way they were going: Helping people instead of just collecting tech (not 100% on the original goal of the brotherhood, but think it was that)
So the outcasts was actually more like the original brotherhood.

Actually, a more accurate statement would be to say that the Outcasts are more like the Mojave Chapter. Which makes sense, as the Mojave BoS is hardline old-style BoS, and the Outcasts left the East Coast BoS for diverging from the "ideal."

Remember, Casdin's main reason for wanting to get in contact with the West Coast isn't to request support, but to tattle on Lyons and see him and those that agree with him shot as traitors. There's a reason a Good Karma character really doesn't need much justification for arriving at Fort Independence and saying "Nice fort. I'll take it."

Tattling on Lyon's isn't the reason, the Elders back east already knew what Lyon's is up to but were split on what to do about it, so they simply decided Elder Lyon's could run his chapter but severed all support.

They do. Outcast is real brotherhood. The Mojave chapter is real brotherhood.
Lyons' chapter is going against the code.

Yup as Peter said the Mojave chapter and outcasts are the real brotherhood. Their goals are to basically hoard old world technology.

Lyon's brotherhood had the same goals but eventually Lyon's felt bad for the Capital Wasteland and wanted to help them. So instead of looking for technology they decided to help out the capital wasteland. The outcasts are a branch of Lyon's brotherhood. They didn't like how he strayed from their goals and left. They then settled down in fort independence and have been trying to contact the main BrotherHood on the west coast

Atleast that's how I think it works

QUOTE: They do. Outcast is real brotherhood. The Mojave chapter is real brotherhood.
Lyons' chapter is going against the code.

Dead on. Lyons' chapter is actually more like the Followers of the Apocalypse with power armor and a big honkin' robot.