Single core baluns do work in the sense that they will pass signals
without strong attenuation in most cases, they will transform
impedances properly, and they are broadband. They will work well in a
system that is already fairly well balanced.

You will need to evaluate your application as single core 4:1 baluns
have the following disadvantages compared to, say, the 4:1 Guanella
two-core current balun:

1. They do not supress common-mode currents well.
2. They require a larger core to work with a given power in a fairly
well-balanced system (a severely unbalanced system might have to be
evaluated more carefully to determine core size).
3. When properly built with a core that couples most all flux to all
windings, the single core balun will have no output into a fully
unbalanced system. An unbalanced system is one where either end of the
load is grounded. A balanced system is one where the load is grounded
in the center. There is also a floating system where the load is not
grounded at all.

They have the advantage that they only use one core.

On an antenna, the effects of using a voltage balun (all single core
4:1 baluns seem to be voltage baluns) instead of a current balun can be
subtle or even undetectable. My experience with an offset fed dipole
was that the antenna receive signal-to-noise ratio degraded under some
conditions without a current balun in place (maybe simply because the
pattern was different). Adding the current balun helped, and the
antenna tuned a little more as I expected it to on the higher bands.
My feedline is near a huge aluminum siding slab, and I think there was
some effect from that.

An offset fed dipole can be pretty unbalanced, though. Other antennas
might do just fine with a single core balun.

I'm sure others will add their comments, and perhaps disagree with me.
This has been a controversial subject in the past, but more recent
discussions have been better.

73,
Glenn Dixon AC7ZN