Piketty's explanation of elasticity of substitution (from his book Capital in the 21st century)

The following is from Thomas Piketty and Gabriel Zucman (2015, From Handbook of Income Distribution, Volume 2, Chapter 15, Part 15.5.3 which is hard to link to directly but get it here):

Take a CES production function $$Y=F(K,L)=(a⋅K^{frac{sigma-1}{sigma}}+(1−a)⋅L^{frac{sigma-1}{sigma}})^{frac{sigma}{1-sigma}}$$

Me: $sigma$ is the elasticity of substitution. They have a small typo here with the outer exponent of $frac{sigma-1}{sigma}$ instead of $frac{sigma}{1-sigma}$, but I'm pretty sure that is wrong so I corrected it here. It doesn't change the rest of what I'm doing.

Back to Piketty and Zucman:

The rate of return is given by $$r = F_{K} = a cdot beta^{frac{-1}{sigma}}$$ with $beta = K/Y$.

The capital share is given by $$ alpha = rcdotbeta = acdotbeta^{frac{sigma-1}{sigma}}$$

Me: Now take the partial derivative of $alpha$ with respect to to $beta$ $$ frac{partial alpha}{partial beta} = a cdot (1-1/sigma) cdot beta ^{-1/sigma}$$

By assumption $a$ is always positive. Because $K$ and $Y$ are always positive so is $beta ^{-1/sigma}$. That means the sign of $alpha$ is the sign of $1-1/sigma$. For $0<sigma<1$ you can see that $1-1/sigma < 0$ and therefore $$frac{partial alpha}{partial beta} < 0$$.