How many spurs?

[AJW]

I have fitted a spur running from a double socket of the main ring, it's for an intergrated cooker hood. I also need to wire in some low voltage under cupboard lights. Can I run another 3 pin socket from the fused spur for these lights?

 

[BR]

No, you cannot do what you have suggested, A spur can only supply one socket outlet. (a double socket outlet is acceptable). Also your socket is most likely protected by a 32A MCB/RCD.
This 32A is two high for 1.5sq lighting cable, Light units, transformers etc. Lights should not be protected by the RCD.
Despite all this, it is bad practice, You can supply lights from a three pin lighting socket but it would be advisable to simply take you power from the nearest ceiling rose. Use 1.5sq twin and earth Cable.

 

[il78]

From this (above) which is in your kitchen now, into (below)

From this you can connect as many sockets as you like as the rating of the fuse is lower than the rating of the cable (2.5mm).
So connect 2.5mm into (below) that will do the hood, then from this..

Into this (below) which will be rated at 5Amp

So you can wire your lights in 1.5mm or suitable rated flex.