Given a choice of a Riemannian metric $g$ on an oriented 4-manifold $M,$ the Hodge star operator $*_g$ is an involution and hence induces a canonical splitting $\Lambda^2(T^*M)= \Lambda^{2,+}(T^*M) \oplus \Lambda^{2,-}(T^*M)$ of the bundle of 2-forms into the self-dual and anti-self-dual 2-forms.
In Claude Lebrun's 1994 ICM address (http://www.mathunion.org/ICM/ICM1994.1/Main/icm1994.1.0498.0507.ocr.pdf) he remarks that this splitting is related to the fact that the adjoint representation of $SO(4)$ on $\frak so(4)$ splits as a direct sum of 3-dimensional representations, thus inducing a Lie algebra isomorphism $\frak so(4)= so(3) \oplus so(3).$
My question: could someone provide more details, or a reference, regarding the relationship between these statements? In particular, how is the Hodge star operator induced by a metric $g$ related to the adjoint representation of $SO(4)$ on $\frak{so}(4)$?
Let $G$ be a Lie group acting on a vector space $V$ with $G$-invariant inner product $\langle \cdot, \cdot \rangle.$ A 2-form $\alpha \in \Lambda^2 V^*$ can be viewed as an element of $\frak{so}$$(V)$ as follows:
There is a natural "raising the index" operation defined by mapping $w \in V^*$ to the unique $v \in V$ such that $\langle v, \cdot \rangle = w.$ We write $v=w^{\#}.$ Now we can use $\alpha$ to define a map $m_{\alpha}: V \to V$ by $ v \mapsto (\alpha(v,-))^{\#}=: \tilde{v}.$ It's easy to see that $\langle m_{\alpha}(v_1), v_2 \rangle= - \langle v_1, m_{\alpha}(v_2) \rangle.$ Hence $m_{\alpha} \in \frak{so}$$(V).$ It's not hard to see that we can also go backwards (i.e. get a 2-form from an element of $\frak{so}$$(V)$). One can finally check that the natural representation of $G$ on $\Lambda^2V^*$ induced by the $G$- action on V induces the adjoint representation on $\frak{so}$$(V).$
Now, specializing to the case $V=\mathbb{R}^4,$ and $G= SO(4)$ and letting $\{e_i\}$ be the standard basis vectors, we have a splitting $\Lambda^2 \mathbb{R}^4= \Lambda^{2,+} \oplus \Lambda^{2,-}$ which is induced by the Hodge star an which splits the standard representation of $SO(4)$ on $\Lambda^2(\mathbb{R}^4).$ Now using the isomorphism of the previous paragraph, we get a splitting of the adjoint representation of $SO(4)$ on $\text{so}_4(\mathbb{R}).$
On the other hand, if we first consider the adjoint representation of $SO(4)$ then there is a distinguished splitting $\text{so}_4(R)= \text{so}_3(R) \oplus \text{so}_3(R).$ One can explicitly check by choosing a basis and following the above isomorphism backwards that this splitting agrees with the splitting induced by the Hodge star.
Finally, following Qiaochu Yuan's suggestion, on can adopt the general "associate G-bundles framework" to globalize the above discussion.