Poincaré algebra and quotient group

[LCSphysicist]

TL;DR Summary

I am having trouble to understand the print below.

I see that the first four equations are definitions. The problem is about the dimensions of the quotient.

Why does the set Kx forms a six dimensional Lie algebra?

 

[Infrared]

There are only $6$ linearly independent vectors $M_{\mu\nu}$ by anti-symmetry and their images in the quotient give a basis.

The quotient is naturally a Lie algebra since $t^4$ is an ideal (by the last two commutation relations).