Investigations into the organisational uptake of artificial intelligence (AI) solutions confirm that there is a growing interest in enterprises and public authorities to use AI. In this context, the lack of understanding of AI concepts in organisations is a significant challenge. As a contribution to addressing this issue, we previously developed and evaluated a morphological box for AI solutions. To further refine this morphological box, the paper follows a well-established scientific method for this purpose: This paper presents the application of a taxonomy development method to our morphological box. We use this method to determine a meta-characteristic, identify the target audience, project the use of the morphological box, and define both subjective and objective ending conditions. We describe several iterations of the development and evaluation loops and present our final results. Our analysis demonstrates the effectiveness of the taxonomy development method in refining and enhancing the morphological box for AI solutions. We further present the application of the morphological box for classifying AI projects with four initial case studies, discuss the results as well as further development directions and potentials of the box.