In trying to understand connection one-forms, I have to learn what a Lie-algebra valued form is. I already understand what a vector-valued form is. I also understand why \nabla (e) = e \otimes \omega where \omega is a one-form, \nabla is an affine connection /covariant differentiation and e is some basis vector. Here \omega (vector) = number But in the case of a connection one-form matrix, I am trying to understand why, when supplied with a "vector", it produces an element of the Lie-algebra. So all of a sudden, it would appear, \omega (vector) = lie-algebra element = element of a vector space! Can someone explain this to me? Many thanks!