  Rotational bands of positive and negative parity are frequently found
experimentally in heavy deformed even-even nuclei along with their
corresponding $E2$ intra-band and $E1$ inter-band transitions. In $SU(3)$
symmetry-based models, calculating the $B(E2)$ strengths is relatively
straightforward due to the relation between the $E2$ operator and the
quadrupole generator of the algebra. However, calculating the $B(E1)$ strengths
is a more difficult task, requiring computations involving the shell-model
expansion of the dipole operator and multi-shell matrix elements. In this work
the $E1$ operator is expanded in $SU(3)$ second quantization and applied to
calculate the inter-band $B(E1)$ strengths of the isotope $^{224}$Th using the
semi-microscopic algebraic quartet model (SAQM) based on the recently proposed
proxy-$SU(3)$ scheme.
