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Example: If ρ is the [[adjoint representation]] of ''G'' on the Lie algebra, then the connection form ω satisfies the first condition (but not the second). The associated [[curvature form]] Ω satisfies both; hence Ω is a tensorial form of adjoint type.
Given ''P'' and ''ρ'' as above one can construct the [[associated vector bundle]] ''E'' = ''P'' ×<sub>''ρ''</sub> ''V''. Tensorial ''q''-forms on ''P'' are in a natural one-to-one correspondence with ''E''-valued ''q''-forms on ''M''. As in the case of the principal bundle F(''E'') above, ''E''-valued forms on ''M'' pull back to ''V''-valued forms on ''P''. Explicitly, given a ''q''-form <math>\overline{\phi}</math> on ''M'', define φ fiberwise by (say at ''u'')
:<math>\phi = u^{-1}\pi^*\overline{\phi}</math>
where ''u'' is viewed as a linear isomorphism <math>V \overset{\simeq}\to E_{\pi(u)} = (\pi^*E)_u</math>. φ is then a tensorial form of type ρ. Conversely, given a tensorial form φ of type ρ, the same formula defines an ''E''-valued form <math>\overline{\phi}</math> on ''M''. In particular, there is a natural bijection
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