Construction of fuzzy S^4

We construct a fuzzy $S^4$, utilizing the fact that ${\mathbf{C}\mathbf{P}}^3$ is an $S^2$ bundle over $S^4$. We find that a fuzzy $S^4$ can be described by a block-diagonal form whose embedding square matrix represents a fuzzy ${\bf CP}^3$\mathrm{.}Wediscusssomependingissuesonfuzzy$S^4$, i.e., precise matrix-function correspondence, associativity of the algebra, and etc. Similarly, we also obtain a fuzzy $S^8$, using the fact that ${\mathbf{C}\mathbf{P}}^7$ is a ${\mathbf{C}\mathbf{P}}^3$ bundle over $S^8$.