We construct combinatorial bases of principal subspaces of standard modules of level k ≥ 1 with highest weight kΛ₀ for the twisted affine Lie algebras of type A_2l-1⁽²⁾, D_l⁽²⁾, E₆⁽²⁾ and D₄⁽³⁾. Using these bases we directly calculate characters of principal subspaces.

We are grateful to Mirko Primc for his useful comments and support. The first named author would also like to thank Drazen Adamovic. The first author is partially supported by the Croatian Science Foundation under the project 2634 and by the QuantiXLie Centre of Excellence, a project cofinanced by the Croatian Government and European Union through the European Regional Development Fund - the Competitiveness and Cohesion Operational Programme (Grant KK.01.1.1.01.0004).