Rotations in $\mathbb{R}^3$ are generated by differential operators $$ O_{yz}=y\frac{d}{dz}-z\frac{d}{dy}, O_{zx}=z\frac{d}{dx}-x\frac{d}{dz}, O_{yx}=y\frac{d}{dx}-x\frac{d}{dy},$$ which satisfy the commutation relations $$[O_{yz},O_{zx}]=O_{yx}$$ and $[O_{ab},O_{bc}]=O_{ac}$ in general, with $O_{ab}=-O_{ba}$.
I suppose rotations in $\mathbb{R}^4$ must be generated by those three generators plus three others $O_{xw}$, $O_{yw}$, $O_{zw}$, defined analogously. This is consistent with the dimension of algebra $so(4)$ being 6.
Now I have seen that $so(4)\sim so(3)\oplus so(3)$, so there should be possible to construct linear combinations of the six $O_{ab}$ generators in $\mathbb{R}^4$ to produce two separate $so(3)$. How is this done?
$$L_{xy}=\tfrac12(O_{xy}+O_{zw}),\quad L_{yz}=\tfrac12(O_{yz}+O_{xw}),\quad L_{zx}=\tfrac12(O_{zx}+O_{yw})$$
$$R_{xy}=\tfrac12(O_{xy}-O_{zw}),\quad R_{yz}=\tfrac12(O_{yz}-O_{xw}),\quad R_{zx}=\tfrac12(O_{zx}-O_{yw})$$