Criteria to find a common non orthonormal basis for two linear operators

Question

I can't find any criteria to determine, in finite dimension, if two operators has a non orthogonal common basis, for example, given two operators A and B if I check

1. $A=A^+$
2. $B=B^+$
3. $[A, B] =0$

In this situation I can affirm that A and B have a common orthonormal basis of eigenvector, or if the 1 and 2 properties are not true for A and B but the third one is, I can say that A and B have only a common eigenvector.

But what properties should I check to know if two operators have a common non orthonormal basis? Can I ask this question to myself or it is incorrect itself?

Often on old tests I find "determine if this two operator have a common eigenvector basis. What kind of basis is this and why?" but I only find on my book criteria to find the orthonormal one.

Thank you and sorry for bad English.