The use of 3D Gaussians as representation of radiance fields has enabled high quality novel view synthesis at real-time rendering speed. However, the choice of optimising the outgoing radiance of each Gaussian independently as spherical harmonics results in unsatisfactory view dependent effects. In response to these limitations, our work, Factorised Tensorial Illumination for 3D Gaussian Splatting or `3iGS' improves upon 3D Gaussian Splatting (3DGS) rendering quality. Instead of optimising a single outgoing radiance parameter, 3iGS enhances 3DGS view dependent effects by expressing the outgoing radiance as a function of a local illumination field and Bidirectional Reflectance Distribution Function (BRDF) features. We optimise a continuous incident illumination field through a Tensorial Factorisation representation, while separately fine-tuning the BRDF features of each 3D Gaussian relative to this illumination field. Our methodology significantly enhances the rendering quality of specular view-dependent effects of 3DGS, while maintaining rapid training and rendering speeds. Code will be released.