Like Columbo, Elementary falls back on the classic motive for murder: greed!
Elementary has an interesting bonus theme. Being in a relationship is work! And it may entail hurting others. And it's work. The money motive consequently is nearly always tied to the underlying emotional motivations I reference in the prior post. The belief that "commitment is only commitment because
it has no expiration date" (from House, spoken by Foreman) can also be attributed to the Elementary writers.
The pilot episode of Elementary presents a husband who seems as irritated by his wife's existence as desirous of her money--referring to himself in the third person, he states that the "husband wants out of the marriage." A later episode "Deja Vu All Over Again" revolves around the same grasp for freedom, only in this case, the husband punishes the wife for the impulse. And the episode "Ears to You" has an ex-wife who couldn't be bothered to go through the hard work of a divorce.
The motivations are mostly all show-not-tell. But the scripts consistently avoid displays of villainous greed--or even need. The murderous significant others are unwilling to give up any of their prerogatives for the sake of the significant other, from money to attention.
Another Season 1 episode "While You Were Sleeping" sets the stage. Sherlock solves the case when he hears a recovering addict admit that when she was on drugs, she only cared about herself, not her doctor boyfriend or what her choices would do to him. He later refers to the murderess--a twin sister killing off heirs to the family money--as having a "little heart."
Sherlock understands the villains' reluctance to work at relationships. He finds them difficult too.
He nevertheless condemns their illegal and harmful solutions. When the supposed activist against the government kills the young woman who helps him, Sherlock shrugs off the "security" implications. He only cares about the man's violent behavior. Like Poirot, he cares most for the individuals "who deserve not to have taken from them their lives." And despite his own reluctance to invest himself in a relationship when he is not sure he has the necessary requisite qualities--especially regarding fathering a child--he expects civilized members of a society to at least try, or, at the very least, not murder.
The theme that relationships take effort underscores the show. Sherlock attempts to build his friendship with Joan. He advises Marcus to take chances in his personal relationships. He encourages Captain Gregson to work at his failing marriage and then helps him in his later relationship with Paige. In the episode "You Do It Yourself," he gives the ultimate British compliment to the young research assistant who agrees to marry the woman he loves, becoming an instant father:
"Good show."