The standard intertwiners form a basis of the principal series endomorphism algebra

For $A=\mathbb{C}[G]$, an $A$-linear map $A{e}_{\chi }\to A{e}_{\chi \mathrm{\prime }}$ is determined by $a=f(e_{\chi })$, which satisfies $e_{\chi }a=a$ and $a{e}_{\chi \mathrm{\prime }}=a$. Conversely each $a\in e_{\chi }A{e}_{\chi \mathrm{\prime }}$ gives $f(y{e}_{\chi })=ya$. Thus this mixed corner is naturally the vector space ${\mathrm{Hom}}_G(I(\chi ),I(\chi \mathrm{\prime }))$ under the models of [F1], and its dimension is $\mathrm{\#}\{w:\chi =w\cdot \chi \mathrm{\prime }\}$ by [F5].

Bruhat decomposition and $e_{\chi }b=\tilde{\chi }(b)e_{\chi }$, $b{e}_{\chi \mathrm{\prime }}=\tilde{\chi }\mathrm{\prime }(b)e_{\chi \mathrm{\prime }}$ show that the mixed corner is spanned by $e_{\chi }\dot{w}{e}_{\chi \mathrm{\prime }}$, one vector per double coset. For $t\in T$, its left character is $\chi (t)$, while moving $t$ across $\dot{w}$ gives character $\chi \mathrm{\prime }({\dot{w}}^{-1}t\dot{w})=(w\cdot \chi \mathrm{\prime })(t)$. Therefore the vector vanishes unless $\chi =w\cdot \chi \mathrm{\prime }$. The remaining family has exactly the dimension computed in step 1.1 and still spans, so it is a basis. Right multiplication gives the corresponding Hom basis with precisely the source and target orientation of step 1.1.

Taking ${\chi }^{\mathrm{\prime }}=\chi$ the surviving indices are exactly $W_{\chi }$ by [F4], so the mixed corner is $e_{\chi }\mathbb{C}[G]e_{\chi }$ with basis $e_{\chi }\dot{w}{e}_{\chi }$, $w\in W_{\chi }$; rescaling each by $q^{\mathrm{\ell }(w)}$ and reindexing $w\mapsto w^{-1}$ (a bijection of $W_{\chi }$) exhibits ${\mathrm{\Theta }}_w$, $w\in W_{\chi }$, as a basis of the corner. By [F2] the right multiplication map $R$ is $\mathbb{C}$-linear and injective from the corner onto ${\mathrm{End}}_{\mathbb{C}[G]}(\mathbb{C}[G]e_{\chi })$, transported to ${\mathrm{End}}_G(I(\chi ))$; hence the $B_w=R_{{\mathrm{\Theta }}_{w^{-1}}}$, $w\in W_{\chi }$, form a $\mathbb{C}$-basis of ${\mathrm{End}}_G(I(\chi ))$. Its cardinality $\mathrm{\mid }{W}_{\chi }\mathrm{\mid }$ agrees with the independent computation ${\dim }_{\mathbb{C}}{\mathrm{End}}_G(I(\chi ))=\mathrm{\mid }{W}_{\chi }\mathrm{\mid }$ of [F6], and the compensation convention of [F2] is exactly what makes each $B_w$ independent of representatives.

Step 2.1 gives the mixed-corner basis indexed by $\{w:\chi =w\cdot {\chi }^{\mathrm{\prime }}\}$ together with the orientation identification of step 1.1, and step 3.1 gives the basis $(B_w{)}_{w\in W_{\chi }}$ of ${\mathrm{End}}_G(I(\chi ))$ with $\mathrm{\mid }{W}_{\chi }\mathrm{\mid }$ elements; all families are finite and the permutation matrices $\dot{w}$ are explicit, so no choice principle is used.