In this paper we generalize the known De Smet, Dillen, Verstraelen and Vrancken (DDVV)-type inequalities for real (skew-)symmetric complex (skew-)Hermitian matrices to arbitrary real, quaternionic matrices. Inspired by ErdÅs-Mordell inequality, establish DDVV-type in subspaces spanned a Clifford system or algebra. We also Böttcher-Wenzel inequality
