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:<math>d_\nabla(\omega\wedge\eta) = d_\nabla\omega\wedge\eta + (-1)^p\,\omega\wedge d\eta</math>
where ω is a ''E''-valued ''p''-form and η is an ordinary ''q''-form. In general, one need not have ''d''<sub>∇</sub><sup>2</sup> = 0. In fact, this happens if and only if the connection ∇ is flat (i.e. has vanishing [[curvature form|curvature]]).
==Lie algebra-valued forms==
An important case of vector-valued differential forms are '''Lie algebra-valued forms'''. These are <math>\mathfrak g</math>-valued forms where <math>\mathfrak g</math> is a [[Lie algebra]]. Such forms have important applications in the theory of [[connection (principal bundle)|connections]] on a [[principal bundle]] as well as in the theory of [[Cartan connection]]s.
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. This operation is denoted by <math>[\omega\wedge\eta]</math> to indicate both operations involved, or often just <math>[\omega, \eta]</math>.
For example, if <math>\omega</math> and <math>\eta</math> are Lie algebra-valued one forms, then one has
:<math>[\omega\wedge\eta](v_1,v_2) = [\omega(v_1),\eta(v_2)] - [\omega(v_2),\eta(v_1)].</math>
With this operation the set of all Lie algebra-valued forms on a manifold ''M'' becomes a [[graded Lie superalgebra]].
The operation <math>[\omega\wedge\eta]</math> can also be defined as the bilinear operation on <math>\Omega(M, \mathfrak g)</math> satisfying
:<math>[(g \otimes \alpha) \wedge (h \otimes \beta)] = [g, h] \otimes (\alpha \wedge \beta)</math>
for all <math>g, h \in \mathfrak g</math> and <math>\alpha, \beta \in \Omega(M, \mathbb R)</math>.
The alternative notation <math>[\omega, \eta]</math>, which resembles a [[Commutator#Ring theory|commutator]], is justified by the fact that if the Lie algebra <math>\mathfrak g</math> is a matrix algebra then <math>[\omega\wedge\eta]</math> is nothing but the [[graded commutator]] of <math>\omega</math> and <math>\eta</math>, i. e. if <math>\omega \in \Omega^p(M, \mathfrak g)</math> and <math>\eta \in \Omega^q(M, \mathfrak g)</math> then
:<math>[\omega\wedge\eta] = \omega\wedge\eta - (-1)^{pq}\eta\wedge\omega,</math>
where <math>\omega \wedge \eta,\ \eta \wedge \omega \in \Omega^{p+q}(M, \mathfrak g)</math> are wedge products formed using the matrix multiplication on <math>\mathfrak g</math>.
==Basic or tensorial forms on principal bundles==
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