How to check whether the production function below is neoclassical or not
C.E.S production functions are indeed tiresome to differentiate, and it is very easy to make a mistake. But hey, this (algebraic manipulation) is part of the job, and one needs to practice moving around symbols , creating common factors, etc. 

But there is another way to show that $F_{KK} &lt;0$ in totally abstract terms.

First, prove 4. (linear homogeneity), it is easy.  

Second, exploit Euler's Homogeneous Function Theorem, and decompose $Y$ using the first derivatives and the variables (this is a very well known result).

Third, re-arrange the previous to obtain an alternative abstract expression of the first derivative of say, capital.

Fourth, differentiate what you got in step Three, to obtain an expression for $F_{KK}$. 

All the above are done using abstract notation $F_K, F_{KK}$,etc and nowhere the actual C.E.S functional form. After step Four, you will need to calculate the cross-partial, which is easy, even for the C.E.S. function.