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Let $M^K_n$ be the moduli space of framed $K$-instantons with instantonnumber $n$ when $K$ is a compact simple Lie group of classical type. Due toDonaldson's theorem, its scheme structure is given by the regular locus of aGIT quotient of $\mu^{-1}(0)$ where $\mu$ is the moment map on the associatedsymplectic vector space of ADHM data.  A main theorem of this paper asserts that $\mu$ is flat for$K=SO(3,\mathbb{R})$ and any $n\ge0$. Hence we complete the interpretation ofthe K-theoretic Nekrasov partition function for the classical groups inNekrasov-Shadchin's work in term of Hilbert series of the instanton modulispaces together with the author's previous results.  We also write ADHM data for the second symmetric and exterior products of theassociated vector bundle of an instanton. This gives an explicitquiver-theoretic description of the isomorphism $M^{K}_{n}\cong M^{K'}_{n}$ forall the pairs $K,K'$ with isomorphic Lie algebras.

The $SO(3)$-instanton moduli space and tensor products of ADHM data