Let $R$ be a commutative ring with unit, let $H\subset G$ be a Lie subgroup, and let
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\begin{align*}
i:BH\to BG
\end{align*}
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be the map of classifying spaces induced by the inclusion $H\hookrightarrow G$. Let $M$ be a space for which principal bundles are classified by homotopy classes of maps into classifying spaces, and let $P\to M$ be a principal $G$-bundle classified by a map
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\begin{align*}
f:M\to BG.
\end{align*}
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Suppose that $P$ admits a reduction of structure group to $H$ classified by a map
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\begin{align*}
g:M\to BH
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such that $f$ is homotopic to $i\circ g$. Then, for every homogeneous universal cohomology class $u\in H^*(BG;R)$,
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\begin{align*}
f^*u=g^*(i^*u)
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in $H^*(M;R)$.
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Consequently, if for a proposed compatible $H$-reduction no classifying map $g:M\to BH$ can make the displayed identity hold for some $u\in H^*(BG;R)$, then $P$ admits no reduction of structure group to $H$ compatible with the chosen classifying data.