One dimension:

Willingness to share Liberties with other people

Far Right = Ayn Rand

Far Left = some guy who just wants to live in a van down by the river and smoke weed all day

As a researcher doing data-stuff: there actually is a somewhat objective way to answer this! I don't know the answer to the question itself though... and the method is quite boring

Usually how data scientists do this is to first collect a bunch of data... let's say we have a 200~300 question comprehensive survey about ppl's political beliefs. This survey would have a dimension of 200-300. We can include all of them but they would offer diminishing information (& is very confusing), so usually people trim it down to the most important dimensions only. We then apply dimensionality reduction/manifold method to reduce highly similar dimensions. I think in social sciences people call this factor analysis. Usually in my field people do PCA followed by UMAP, social scientists I think may do something differently but PCA is quite universal

Then researchers will be able to tell a few mathematically identified dimensions that contribute the most to the results. Say if the first dimension contributes 70% of the variation of people's differences, and the second dimension another 25%... then we would have a 2-dimension model that can explain 95% of the differences and would be good enough. If the first dimension only 10%, second 8%... then a good model will need a lot more dimensions. This doesn't tell what the dimensions are though, that's up to the researchers to identify. If all of these work well, we'd have a simple, N-dimension model suggesting how people's political beliefs are... and some of these might not map to what people would intuitively think of

Unless I'm mistaken, Big Five personality traits is developed this way for example... About politics, I found a 2013 research article that suggested two political dimensions: economic and social ideology

I guess this doesn't quite answer the question... it just states how political dimensions (or any dimensions in data fields, really) came from, and the fact that there's an old paper suggesting a two factor model of economic + social ideology. I don't know how many dimensions are sufficient for politics, not to count for the fact that different countries/cultures treat this differently

I'd say a 3D one is good for most cases. Personal freedom, economic freedom, and political freedom, AKA the NationStates ~~Non~~-Political compass.

Functionally, though, while it's an easy visual I don't necessarily know how useful political compasses are in general. One of my main gripes with the currently popular one is that it's personal+economic freedom, and I don't think there should be an economy at all[1]. I end up very lib left on it (something like -8,-8 or -9,-9) despite, again, not thinking economy is something that should even exist. Would that lean me more right on the economy, or left? Who's to say! You're not economically free if there is none, but also you're as economically free as you want to be!

1: My political ideals are not how I vote, so I go as close as I functionally can.

How about something about supporting the people who are poor, unemployed, or sick? Socialists and leftists love it, while right-wing capitalists hate it. I don’t know how to phrase it concisely like you did.

And then there's also the classic socialist-capitalist debate about worker rights. So employee rights could be another axis. Should the owner just exploit the workers or do the workers get to have weekends off, 8 h workdays, various vacations, safe working environment, fair wages etc.

Oh, and then there are various language based divisions too. There are entire parties dedicated to supporting specific language groups, but I guess you could summarize it as "support of minority language groups". How about just lumping all minority groups into a single axis? How about something like: minority rights vs. majority? Nowadays, that includes sexual and gender minorities too.

How about city vs. rural life? Not too many decades ago, farming was a big part of life, so there were also many farmers who voted. Hence, we had farmer parties, and we still do to some extent. Now that farming is mostly automated, not that many voters care about farming. It's just another industry, just like steel, paper or electronics. Historically speaking, city. vs rural life was definitely a political axis. Nowadays, not so much.

In any case, it's a really complicated topic, so we're going to need a lot of dimensions. Your suggestions are a good start. Just add a few more, and eventually you have enough. If you're really technical about it, each and every question is a new dimension, but if you should group them together into broader topics. That way, you're definitely going to end up in at least 20 dimensions.

IMHO is more like a political 3D spectrum between globalism and nationalism.