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.

A Lie-algebra-valued differential ${\displaystyle k}$-form on a manifold, ${\displaystyle M}$, is a smooth section of the bundle ${\displaystyle ({\mathfrak {g}}\times M)\otimes \wedge ^{k}T^{*}M}$, where ${\displaystyle {\mathfrak {g}}}$ is a Lie algebra, ${\displaystyle T^{*}M}$ is the cotangent bundle of ${\displaystyle M}$ and ${\displaystyle \wedge ^{k}}$ denotes the ${\displaystyle k^{\text{th}}}$ exterior power.

The wedge product of ordinary, real-valued differential forms is defined using multiplication of real numbers. For a pair of Lie algebra–valued differential forms, the wedge product can be defined similarly, but substituting the bilinear Lie bracket operation, to obtain another Lie algebra–valued form. For a ${\displaystyle {\mathfrak {g}}}$-valued ${\displaystyle p}$-form ${\displaystyle \omega }$ and a ${\displaystyle {\mathfrak {g}}}$-valued ${\displaystyle q}$-form ${\displaystyle \eta }$, their wedge product ${\displaystyle [\omega \wedge \eta ]}$ is given by

where the ${\displaystyle v_{i}}$'s are tangent vectors. The notation is meant to indicate both operations involved. For example, if ${\displaystyle \omega }$ and ${\displaystyle \eta }$ are Lie-algebra-valued one forms, then one has

The operation ${\displaystyle [\omega \wedge \eta ]}$ can also be defined as the bilinear operation on ${\displaystyle \Omega (M,{\mathfrak {g}})}$ satisfying

for all ${\displaystyle g,h\in {\mathfrak {g}}}$ and ${\displaystyle \alpha ,\beta \in \Omega (M,\mathbb {R} )}$.

Some authors have used the notation ${\displaystyle [\omega ,\eta ]}$ instead of ${\displaystyle [\omega \wedge \eta ]}$. The notation ${\displaystyle [\omega ,\eta ]}$, which resembles a commutator, is justified by the fact that if the Lie algebra ${\displaystyle {\mathfrak {g}}}$ is a matrix algebra then ${\displaystyle [\omega \wedge \eta ]}$ is nothing but the graded commutator of ${\displaystyle \omega }$ and ${\displaystyle \eta }$, i. e. if ${\displaystyle \omega \in \Omega ^{p}(M,{\mathfrak {g}})}$ and ${\displaystyle \eta \in \Omega ^{q}(M,{\mathfrak {g}})}$ then

where ${\displaystyle \omega \wedge \eta ,\ \eta \wedge \omega \in \Omega ^{p+q}(M,{\mathfrak {g}})}$ are wedge products formed using the matrix multiplication on ${\displaystyle {\mathfrak {g}}}$.