Oxaloacetate contributes to fatty acid synthesis, but it’s not a lipid, because, among other reasons, it’s not hydrophobic. This is also why it cannot cross mitochondrial membranes. Glucose can be synthesized from glycerol, but this process occurs via dihydroxyacetone phosphate, and doesn’t involve oxaloacetate. Oxaloacetate is synthesized from pyruvate, which is the end product of glycolysis, so oxaloacetate cannot be an intermediary in glycolysis. However, lactate can be converted to pyruvate, which is the principle substrate for gluconeogenesis (sometimes called “reverse glycolysis”). In gluconeogenesis, oxaloacetate is an intermediary in the conversion of pyruvate to phosphoenolpyruvate, and so that makes it an intermediary, too, in the synthesis of glucose from lactate.