I must plead extreme busy-ness. Ok, I expect we will get to that in a minute.
Lie group is scale the size of everything up or down, uniformly.
Where one generally says something like 'um'. This operation is in fact linear on the Lie algebra g. I'm sure you are right, only I haven't got there yet. The possible solutions form a vector space, and the rotation operation is a linear operator on this vector space. That's the whole idea of a representation, if you include the stuff Toby mentioned above.