I am asking mainly for $\psi$ , how we use this to differentiate L .Since $\psi $ is a $4\times 1$ matrix.

@ Rob Tan , But how your answer shows that general Lorentz matrix can uniquely decomposed into pure rotation and pure boost?Here it only shows that Lorentz transformation has 6 generators.

Nice argument. I got (GL)^T= - GL. by expandindg" exp(L)" and '" exp(L^T) " in terms of "L" and " L^(T)", in generic Lorentz transformation condition, then comparing linear terms in "L" and " L^(T)" on both side of the equation.Then all other higher terms cancel with each other due to the obtained condition .Is this the correct way ?

How? Its determinent is zero.