  Stars with an initial mass of $\sim 7$-11 solar masses form degenerate
oxygen-neon cores following carbon burning. Electron captures in such cores can
trigger runaway oxygen burning, resulting in either a collapse or a
thermonuclear explosion. We provide a detailed description of the formalism
used in previous work and apply it to two further forbidden transitions that
are relevant to degenerate oxygen-neon cores: the
$4^+_\text{g.s.}\rightarrow{}2^+_1$ transition in
$^{24}\text{Na}(e^-,\nu_e)^{24}\text{Ne}$ and the ${5/2^+_{\text{g.s.}}
\rightarrow{} 1/2^+_{\text{g.s.}}}$ transition in
$^{27}\text{Al}(e^-,\nu_e)^{27}\text{Mg}$. The relevant nuclear matrix elements
are determined through shell model calculations and constraints from CVC
theory. We then investigate the astrophysical impact using the stellar
evolution code MESA and through timescale arguments. In the relevant
temperature range, the forbidden transitions substantially reduce the threshold
densities for $^{24}\text{Na}(e^-,\nu_e)^{24}\text{Ne}$ and
$^{27}\text{Al}(e^-,\nu_e)^{27}\text{Mg}$. In the MESA models,
$^{24}\text{Na}(e^-,\nu_e)^{24}\text{Ne}$ now occurs immediately following the
onset of $^{24}\text{Mg}(e^-,\nu_e)^{24}\text{Na}$. The impact on the overall
evolution is uncertain: this is due to known difficulties in accounting for
convective instabilities triggered by the $A=24$ electron captures. The
transition between $^{27}\text{Al}$ and $^{27}\text{Mg}$ may have a minor
effect on the early evolution but is unlikely to affect the outcome. The
studied transitions should be included when calculating weak interaction rates
between $^{24}$Na and $^{24}$Ne for temperatures
$\log_{10}(T[\text{K}])\lesssim8.5$ and between $^{27}$Al and $^{27}$Mg for
$\log_{10}(T[\text{K}])\lesssim8.8$.
