Killing tensor notation quick questions

[binbagsss]

My notes read that, the quantity $K^{2}=K_{uv}V^{u}V^{v}$ is constant along geodesics, where $K$ is a killing vector. I know my definition that the quantity on the RHS is conserved, I'm just wondering why do we call it $K^{2}$, rather than anything else?

In analogy to a killing vector, if $K$ was a killing vector, would we instead call this $L$ or whatver, but without a squared sign?

Thanks in advance for any guidance, or a point towards some notes on this.

 

[binbagsss]

Bump. Also by calling it $K^{2}$ aren't we asumming its positive, how would we know this? thank you.