 Solution

Applying $P[a,b,c]=[b,a,b+c]$ to the eight weights of $d_3$ givesThis is exactly the union of the weight sets of $d_1$ and $d_2$. It expresses the branching rule for the standard inclusion $D_3\cong A_3\subset B_3$: the eight-dimensional spinor representation of ${\mathfrak{so}}_7$ restricts to the two four-dimensional chiral spinor representations of ${\mathfrak{so}}_6\cong {\mathfrak{sl}}_4$,