Sterilize jars by boiling them and place small plates in the freezer for testing the jam later.
Wash kumquats thoroughly, cut off stems and bottoms, and slice into thin rounds, removing seeds carefully.
Place seeds in a small container with 4 tablespoons of water, cover, and refrigerate.
Combine sliced kumquats and sugar in a pot, mix lightly, cover, and let stand for half a day.
Bring the seed and water mixture to a boil and simmer for 5 minutes.
Add 1 cup of water to the kumquats and bring to a boil, adjusting heat to prevent burning. Strain the seed water and add it to the pot, along with optional lemon juice.
Reduce heat to a gentle simmer, skim any foam, and avoid stirring too much. Simmer for about 20 minutes until thickened.
Test consistency by spooning jam onto a chilled plate; if it sets after 1 minute, turn off the heat.
Pour hot jam into sterilized jars and loosely close the lids.
Place jars in a pot with hot water halfway up, cover, and heat on low to medium for 15 minutes.
Remove jars, tighten lids immediately, and allow to cool.
Check seals; properly sealed jars can be stored at room temperature for up to 2 months. Refrigerate after opening.