Neutral technical progress

Recently I learnt that Cobb-Douglas production function has elasticity of substitution equal to 1, therefore it has neutral technical progress. Then Leontief production function has zero elasticity of substitution. Does that imply that it cannot have neutral technical progress? On the other hand, factors of production in Leontief cannot be substituted for one another so I thought it can never have labour or capital saving technical progress.

There are different notions of neutrality of technical progress in macroeocnomics.

You can have Hicks-neutral technical progress - that is a technical progress that increases the marginal productivity of all factors of production by the same proportion at the same capital-labor ratio. An example of Hicks-neutral production function would be one where $$Y^*= F(AK, A L) = A F( K,L)$$ and consequently $$\frac{\partial Y / \partial K}{\partial Y / \partial K}= \frac{AF_k}{AF_L}= \frac{F_K}{F_L}$$. Also note the technical progress would be neutral when its neither capital or labor saving, according to Hick's definition in his the theory of wages, labor saving technology would be one that improves the marginal product of labor relatively more and capital saving that which improves the marginal product of capital relative to labor.

I dont recall ever seeing Leontief production function that would be assumed to be Hicks-neutral in research papers but I think it should be possible to have special cases such as $$F = \min{ \{A K, A L}\}$$ where $$K$$ and $$L$$ have to be consumed exactly in the same proportion you could say that the technology is Hicks-neutral as it is neither labor nor capital saving, also the $$K$$ and $$L$$ ratio would have to remain constant as they are perfect complements and here we would need one unit of capital for every one unit of labor.

Whether the production function is Hicks-neutral depends also on factor elasticity of substitution as well as product elasticity of demand but that is more in sense that the same production function can be or not be Hicks-netural depending on the elasticity of substitution it has.

Then you can also have Harrod-neutral technical progress. In this definition the technological change is neutral as long as capital-output ratio and the rate of profit stays constant. An example of such function would be a function where technology is labor augmenting like $$F(K, AL)$$.

Next, you can also have Solow-neutral technical progress which is defined as technological change that leaves the relative input shares unchanged for a given labour-output ratio. With example being capital augmenting technology $$F(AK,L)$$.

Moreover, you can even have Leontief-neutral technical change. This occurs when the augmentation effect of technology would be different for both factors of production so you would have $$F(A_1 K, A_2L)$$ where $$A_1\neq A_2$$.

None of the above definitions seem to exclude Leontief production function per se.