Big "Here's how I'd do it" post bellow the fold:
If possible, I recommend gutting the wiring and doing your own to keep it simple, AND make sure you get it correct. Get some lamp cord, 16 gauge stranded wiring in white and black, and get some connectors that can handle multiple wires (conductors) at a time. Make sure you have a way to pull pairs of wires through the lamp body, too. You may be able to tie a pull-string to the ends of your old wiring and use it to pull the new wiring through (common technique done when installing network wiring in a building).
Wire the sockets in parallel. That means splitting the hot and the neutral into two branches, each branch going to a lamp socket. To steal someone else's picture:
Here's my rough rules-of-thumb, specific to US implementations (not familiar with other countries' conventions):
Hot connects to the smooth side of the lamp cord, with the skinnier blade of the plug if it has one. On the lamp socket, it connects to the brass-colored screw on the lamp socket.
Neutral is the ribbed side of the lamp cord (if present), and attaches to the gray-colored screw in the lamp socket and the fat blade on the plug.
But with a lot of older appliances and lamps, the plugs aren't polarized and the wiring isn't color-coded or smooth/ribbed differentiated. You can safely replace all that wiring if you want, from the plug to the sockets. If you do your own wiring on the inside, use black for hot/positive and white for neutral/negative.
Just remember that black = hot = brass = skinny and white = neutral = gray = fat.
If you have the room to work with them, you want mechanical wire connects for things that are going on mains power even if they're only powering commodity LED bulbs. Don't solder if you can help it, it makes everything more work.
I vastly prefer lever nuts over other types of connectors because they're simple, fast, and reversible. They typically come in 2-conductor, 3-conductor, and 5-conductor varieties. The go-to brand is WAGO, but Ideal also makes perfectly good ones that might be more available locally.
If you're only branching the incoming power 2 ways (e.g. to two sockets), you just need some 3-conductor lever nuts. If your socket fixture has pigtails (attached wires to make connections from instead of screw terminals) you might want some 2-conductor or inline butt connectors also.
If you can't pull the old wiring, you may be able to trace it with a multimeter in "continuity" mode. The center (at the inside bottom) of a lamp socket should be the "hot" wire and the metal screw collar the bulb goes into should be the "neutral." If you use a multimeter and put one probe on the center middle of the socket, try touching the other probe to the other end of the wires and see which one gives you a beep. Label that end as your hot. Etc.
Wire the sockets in parallel. That means splitting the hot and the neutral into two branches, each branch going to a lamp socket. To steal someone else's picture:
Here's my rough rules-of-thumb, specific to US implementations (not familiar with other countries' conventions):
Hot connects to the smooth side of the lamp cord, with the skinnier blade of the plug if it has one. On the lamp socket, it connects to the brass-colored screw on the lamp socket.
Neutral is the ribbed side of the lamp cord (if present), and attaches to the gray-colored screw in the lamp socket and the fat blade on the plug.
But with a lot of older appliances and lamps, the plugs aren't polarized and the wiring isn't color-coded or smooth/ribbed differentiated. You can safely replace all that wiring if you want, from the plug to the sockets. If you do your own wiring on the inside, use black for hot/positive and white for neutral/negative.
Just remember that black = hot = brass = skinny and white = neutral = gray = fat.
If you have the room to work with them, you want mechanical wire connects for things that are going on mains power even if they're only powering commodity LED bulbs. Don't solder if you can help it, it makes everything more work.
I vastly prefer lever nuts over other types of connectors because they're simple, fast, and reversible. They typically come in 2-conductor, 3-conductor, and 5-conductor varieties. The go-to brand is WAGO, but Ideal also makes perfectly good ones that might be more available locally.
If you're only branching the incoming power 2 ways (e.g. to two sockets), you just need some 3-conductor lever nuts. If your socket fixture has pigtails (attached wires to make connections from instead of screw terminals) you might want some 2-conductor or inline butt connectors also.
If you can't pull the old wiring, you may be able to trace it with a multimeter in "continuity" mode. The center (at the inside bottom) of a lamp socket should be the "hot" wire and the metal screw collar the bulb goes into should be the "neutral." If you use a multimeter and put one probe on the center middle of the socket, try touching the other probe to the other end of the wires and see which one gives you a beep. Label that end as your hot. Etc.
