Jesus said, "If you love me you will obey my commandments" and "If anyone loves me, he will keep my word." These are not conditional statements that are placing a demand. It does not mean "your obedience proves your love." It is a promise. It means "love will cause you to obey."
If you love your wife you won't leave her. However, the fact that you don't leave her doesn't mean that you love her. You don't prove your loving by just hanging around. But your staying with her is motivated by love. It is because of love that you stay with your wife. Likewise it is because of love that we obey God's commandments. You can however keep the rules and not even know God just like you can stay with your wife and not show her one bit of love. Obedience is just a fruit, an outward evidence, of inward love.
Obedience does not produce love. Love does produce obedience and it is God's love for us that produces our love for Him!
Love only exists in a relationship. Doing can be an expression of love but love comes from knowing not from doing.
The word Jesus uses for love describes a self-giving love. You only give yourself to someone you trust. The more you trust the more you are willing to give and share. Trust, or faith as it is often called in the Bible, is what opens the way to the growth of knowing, and loving, and giving. Trust means you consider God to be trustworthy enough to know, that He is truthful about Himself, and loving enough to be encountered.
Jesus commanded us to "love one another just as He has loved us." As John later wrote "We love because God first loved us." It is His love that shows us how to love. Jesus is essentially saying "I will love you so much that you will love one another, obey my commands, and my love will for you makes us friends. I will fulfill the requirement of loving. I chose you and chose to love you so much that you will love me and love others. I will be loving enough for the both of us. You will love because of me and you will love with my very own love."
"If you love me you will obey my commandments" (John 14:15)
"If anyone loves Me, he will keep My word" (John 14:23)
"This is My commandment, that you love one another, just as I loved you." (John 15:12)
"We love, because He first loved us" (1 John 4:19)