Endomorphism-valued exterior product ($(\alpha \otimes A)\wedge (\beta \otimes B)=(\alpha \wedge \beta )\otimes AB$)

Exterior product of differential forms combined with endomorphism composition is associative and generally noncommutative. For an endomorphism-valued one-form $a$, $(a\wedge a)(u,v)=[a(u),a(v)]$. The endomorphism bundle connection extends as a graded derivation of this algebra.