Elaborating on T.P. Ho's correct answer:

1. Given a massive external on-shell particle (an electron) with a non-zero time-like 4-momentum $p^{\prime}\neq 0$ (we can e.g. pick a rest frame where the $3$-momentum ${\bf p}^{\prime}={\bf 0}$ is zero), and

2. given a massless external on-shell particle (a photon) with a light-like 4-momentum $k^{\prime}$,

3. then the denominator of the virtual electron propagator is $$\begin{align} (p^{\prime}+k^{\prime})^2-m^2~=~&(p^{\prime 2}-m^2)+ k^{\prime 2} +2p^{\prime}\cdot k^{\prime}\cr ~=~&0+0+2p^{\prime}\cdot k^{\prime},\end{align}\tag{1} $$ using the $(+,-,-,-)$ Minkowski sign convention.

4. The virtual electron propagator (1) can be on-shell iff the inner product $$p^{\prime}\cdot k^{\prime}~=~0\tag{2}$$ vanishes.

5. Eq. (2) implies that the photon is soft $$k^{\prime}~=~0,\tag{3}$$ which is a frame-independent statement. (To prove eq. (3) use e.g. the rest frame for $p^{\prime}$.)

Elaborating on T.P. Ho's correct answer:

1. Given a massive external on-shell particle (an electron) with a non-zero time-like 4-momentum $p^{\prime}\neq 0$ (we can e.g. pick a rest frame where the $3$-momentum ${\bf p}^{\prime}={\bf 0}$ is zero), and

2. given a massless external on-shell particle (a photon) with a light-like 4-momentum $k^{\prime}$,

3. then the denominator of the virtual electron propagator is $$\begin{align} (p^{\prime}+k^{\prime})^2\pm m^2~=~&(p^{\prime 2}\pm m^2)+ k^{\prime 2} +2p^{\prime}\cdot k^{\prime}\cr ~=~&0+0+2p^{\prime}\cdot k^{\prime},\end{align}\tag{1} $$ using the $(\mp,\pm,\pm,\pm)$ Minkowski sign convention, respectively.

4. The virtual electron propagator (1) can be on-shell iff the inner product $$p^{\prime}\cdot k^{\prime}~=~0\tag{2}$$ vanishes.

5. Eq. (2) implies that the photon is soft $$k^{\prime}~=~0,\tag{3}$$ which is a frame-independent statement. (To prove eq. (3) use e.g. the rest frame for $p^{\prime}$.)

Elaborating on T.P. Ho's correct answer:

1. Given a massive external on-shell particle (an electron) with a non-zero time-like 4-momentum $p^{\prime}\neq 0$ (we can e.g. pick a rest frame where the $3$-momentum ${\bf p}^{\prime}={\bf 0}$ is zero), and

2. given a massless external on-shell particle (a photon) with a light-like 4-momentum $k^{\prime}$,

3. then the denominator of the virtual electron propagator is $$ (p^{\prime}+k^{\prime})^2-m^2~=~(p^{\prime 2}-m^2)+ k^{\prime 2} +2p^{\prime}\cdot k^{\prime}~=~2p^{\prime}\cdot k^{\prime}.\tag{1} $$

4. The virtual electron propagator can be on-shell (1) iff the inner product $$p^{\prime}\cdot k^{\prime}~=~0\tag{2}$$ vanishes.

5. Eq. (2) implies that the photon is soft $$k^{\prime}~=~0,\tag{3}$$ which is a frame-independent statement. (To prove eq. (3) use e.g. the rest frame for $p^{\prime}$.)

Elaborating on T.P. Ho's correct answer:

1. Given a massive external on-shell particle (an electron) with a non-zero time-like 4-momentum $p^{\prime}\neq 0$ (we can e.g. pick a rest frame where the $3$-momentum ${\bf p}^{\prime}={\bf 0}$ is zero), and

2. given a massless external on-shell particle (a photon) with a light-like 4-momentum $k^{\prime}$,

3. then the denominator of the virtual electron propagator is $$\begin{align} (p^{\prime}+k^{\prime})^2-m^2~=~&(p^{\prime 2}-m^2)+ k^{\prime 2} +2p^{\prime}\cdot k^{\prime}\cr ~=~&0+0+2p^{\prime}\cdot k^{\prime},\end{align}\tag{1} $$ using the $(+,-,-,-)$ Minkowski sign convention.

4. The virtual electron propagator (1) can be on-shell iff the inner product $$p^{\prime}\cdot k^{\prime}~=~0\tag{2}$$ vanishes.

5. Eq. (2) implies that the photon is soft $$k^{\prime}~=~0,\tag{3}$$ which is a frame-independent statement. (To prove eq. (3) use e.g. the rest frame for $p^{\prime}$.)