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We study intermediate--z SNe Ia using the empirical physical diagrams whichenable to learn about those SNe explosions. This information can be very usefulto reduce systematic uncertainties of the Hubble diagram of SNe Ia up to highz. The study of the expansion velocities and the measurement of the ratio$\mathcal{R}$(\SiII) allow to subtype those SNe Ia as done for nearby samples.The evolution of this ratio as seen in the diagram $\mathcal{R}$(\SiII)--(t)together with $\mathcal{R}$(\SiII)$_{max}$ versus (B-V)$_{0}$ indicateconsistency of the properties at intermediate z compared with local SNe. Atintermediate--z, the expansion velocities of Ca II and Si II are similar to thenearby counterparts. This is found in a sample of 6 SNe Ia in the range0.033$\leq z \leq$0.329 discovered within the {\it International TimeProgramme} (ITP) of {\it Cosmology and Physics with SNe Ia} during the springof 2002. Those supernovae were identified using the 4.2m William HerschelTelescope. Two SNe Ia at intermediate z were of the cool FAINT type, one beinga SN1986G--like object highly reddened. The $\mathcal{R}$(\SiII) ratio as wellas subclassification of the SNe Ia beyond templates help to place SNe Ia intheir sequence of brightness and to distinguish between reddened andintrinsically red supernovae. This test can be done with very high z SNe Ia andit will help to reduce systematic uncertainties due to extinction by dust. Itshould allow to map the high-z sample into the nearby one.Comment: 55 pages, 22 figures, submitted to The Astrophysical Journal (figures  reduced for astro-ph

TYPE Ia SNe ALONG REDSHIFT: THE $\mathcal {R}$(Si II) RATIO AND THE EXPANSION VELOCITIES IN INTERMEDIATE-zSUPERNOVAE