Lie algebra-valued differential form

In differential geometry, a Lie algebra-valued form is a differential form with values in a Lie algebra. Such forms have important applications in the theory of connections on a principal bundle as well as in the theory of Cartan connections.
Formal Definition.
A Lie algebra-valued differential "k"-form on a manifold, formula_1, is a smooth section of the bundle formula_2, where formula_3 is a Lie algebra, formula_4 is the cotangent bundle of formula_1 and Λ"k" denotes the "k"th exterior power.
Wedge product.
Since every Lie algebra has a bilinear Lie bracket operation, the wedge product of two Lie algebra-valued forms can be composed with the bracket operation to obtain another Lie algebra-valued form. For a formula_3-valued "p"-form formula_7 and a formula_3-valued "q"-form formula_9, their wedge product formula_10 is given by 
where the "v""i"'s are tangent vectors. The notation is meant to indicate both operations involved. For example, if formula_7 and formula_9 are Lie algebra-valued one forms, then one has
The operation formula_10 can also be defined as the bilinear operation on formula_16 satisfying
for all formula_18 and formula_19.
Some authors have used the notation formula_20 instead of formula_10. The notation formula_20, which resembles a commutator, is justified by the fact that if the Lie algebra formula_23 is a matrix algebra then formula_10 is nothing but the graded commutator of formula_7 and formula_9, i. e. if formula_27 and formula_28 then
where formula_30 are wedge products formed using the matrix multiplication on formula_23.
Operations.
Let formula_32 be a Lie algebra homomorphism. If φ is a formula_3-valued form on a manifold, then "f"(φ) is an formula_34-valued form on the same manifold obtained by applying "f" to the values of φ: formula_35.
Similarly, if "f" is a multilinear functional on formula_36, then one puts
where "q" = "q"1 + … + "q""k" and φ"i" are formula_3-valued "q""i"-forms. Moreover, given a vector space "V", the same formula can be used to define the "V"-valued form formula_39 when
is a multilinear map, φ is a formula_3-valued form and η is a "V"-valued form. Note that, when
giving "f" amounts to giving an action of formula_3 on "V"; i.e., "f" determines the representation
and, conversely, any representation ρ determines "f" with the condition (*). For example, if formula_44 (the bracket of formula_3), then we recover the definition of formula_46 given above, with ρ = ad, the adjoint representation. (Note the relation between "f" and ρ above is thus like the relation between a bracket and ad.) 
In general, if "α" is a formula_47-valued "p"-form and φ is a "V"-valued "q"-form, then one more commonly writes α⋅φ = "f"(α, φ) when Explicitly,
With this notation, one has for example:
Example: If ω is a formula_3-valued one-form (for example, a connection form), ρ a representation of formula_3 on a vector space "V" and φ a "V"-valued zero-form, then
Forms with values in an adjoint bundle.
Let "P" be a smooth principal bundle with structure group "G" and formula_53. "G" acts on formula_3 via adjoint representation and so one can form the associated bundle:
Any formula_56-valued forms on the base space of "P" are in a natural one-to-one correspondence with any tensorial forms on "P" of adjoint type.