In Atiyah's famous paper "Bott periodicity and the index of elliptic operators" section 4, he proved the splitting principle for unitary groups (Propostion 4.9 in that paper), namely:

Let $j: T\rightarrow U$ be the inclusion of the maximal torus in the unitary group $U=U(n)$. Fro any compact $U$-space $X$ let $$ j^*:K_U(X)\rightarrow K_T(X) $$ be the restriction map. Then $j^* $ has a (functorial) left inverse $$ j_*:K_T(X)\rightarrow K_U(X). $$

The construction of $j_*$ is done by repeatedly using the Dolbeault operator on projective spaces.

In a following remark Atiyah wrote: "However we can equally well define it at one go by using the Dolbeault complex of the flag manifold $U/T$. The important point is that the sheaf cohomology of $U/T$—like that of any rational variety—has the same properties as for projective space. Thus, more generally, we can replace $U$ in (4.9) by any compact connected Lie group: it being well known that $G/T$ has the structure of a homogeneous rational algebraic variety."

My question is: has the proof using the flag manifold been written down? Are there any references?