The present study develops the Clifford algebra Cl-5,Cl-0 within a dermatological task to diagnose skin melanoma using images of skin lesions, which are modeled here by means of 5D lesion feature vectors (LFVs). The LFV is a numerical approximation of the most used clinical rule for melanoma diagnosis - ABCD. To generate the Cl-5,Cl-0 we develop a new formula that uses the entries of a 5D vector to calculate the entries of a 32D multivector. This vector provides a natural mapping of the original 5D vector onto the 2-, 3-, 4-vector Cl-5,Cl-0 subspaces. We use a sample set of 112 5D LFVs and apply the new formula to calculate 112 32D multivectors in the Cl-5,Cl-0. Next we map the 5D LFVs onto the 2-, 3-, 4-vector subspaces of the Cl-5,Cl-0. In every subspace we apply a binary support vector machine to classify the mapped 112 LFVs. With the obtained results we calculate six metrics and evaluate the effectiveness of the diagnosis in every subspace. At the end of the paper we compare the classification results, obtained in every subspace, with the results obtained by the four diagnosing rules most used in clinical practice and contemporary machine learning methods. This way we reveal the potential of using Clifford algebras in the analysis and classification of medical images.