The torsion index of a spin group Spin(d), describing the splitting behaviour of generic Spin(d)-torsor E, is a 2-power 2^t with the torsion exponent t determined by B. Totaro in 2005. The critical exponent i_t is responsible for partial splitting behaviour of E and takes values inside the doubleton {t-1, t}. For all d less than or equal to 16, the value of i_t is known to be high. The very first case of the low value, obtained very recently, is d=17. In the present work, we develop a new method which allows one to show that i_t=t-1 for most d. In particular, it is shown that i_t is low for every d=2^r+1 with r greater than or equal to 4 as well as for d=2023, playing the role of a "randomly chosen" high dimension. For d=2023, using an extension of the new method (applicable to arbitrary d), several exponents beyond the critical one are also determined.

The author's work has been supported by a Discovery Grant from the National Science and Engineering Research Council of Canada. It has been finalized during his stay at the Universite de Lorraine.