For reference, here are links about digital filters:
Chris Wipf implementations:
Perform IIR filtering sample by sample on TYPE (float/double/long double). /// Implements cascaded direct form II second order sections.
https://git.ligo.org/-/snippets/137
Python implementations of second order sections filtering (scipy.signal)
https://github.com/scipy/scipy/blob/v1.13.0/scipy/signal/_signaltools.py#L4245-L4354
Transposed direct form 2 Direct form II of this Wikipedia page:
https://en.wikipedia.org/wiki/Digital_biquad_filter
https://github.com/scipy/scipy/blob/v1.13.0/scipy/signal/_sosfilt.pyx
sosfilt.py line 75 has the comment “Use direct II transposed structure”