Do what should be done with sincerity, and do not take ownership of the results.

"Your rights are in action, not in results." This is a well-known phrase from the Bhagavad Gita. You can choose to perform the duties assigned to you, but you cannot completely control the outcome.

This does not mean that you should act carelessly without considering the consequences. What you need to let go of is the attachment to the outcome, not the sincerity of your actions. Furthermore, do not think that only you are responsible for the results or the cause, and do not resort to inaction by saying "if I don't hold onto the fruit, then nothing will happen." The path shown here is one where you exhaust all efforts while letting go of attachment to both the outcome (fruit) and the sense of being an actor.

Japanese Translation from the Gita Press Edition - Chapter 2, Section 47: "The Right to Action, Non-Attachment to Results."

Having described the glory and fruits of Karma Yoga as expressed in equanimity, the Lord will now explain the nature of Karma Yoga in two sections, and encourage Arjuna to properly establish himself in Karma Yoga and perform his duty.

Sanskrit original text

कर्मण्येवाधिकारस्ते मा फलेषु कदाचन ।
मा कर्मफलहेतुर्भूर्मा ते सङ्गोऽस्त्वकर्मणि ॥ ४७ ॥

Romanization

karmaṇy evādhikāras te mā phaleṣu kadācana |
mā karma-phala-hetur bhūr mā te saṅgo 'stv akarmaṇi || 47 ||

Your rights exist only in action, and never in the results. Do not be a cause for producing the fruits of actions. Also, do not have an attachment to non-action. (47)

In this section, "Karma" refers to a person's Varna (social class), their position in society, Ashrama (stage of life), and the duties prescribed according to their nature and circumstances.

Sinful acts prohibited by scriptures do not fall within this scope. A person is not permitted to perform them; they are only performed out of impulsive desires, and such actions are not recognized for that person. Therefore, those who commit such acts will be punished through suffering and torment in hells, among other things.

In this section, when the subject says "Your rights exist only in actions," they are trying to convey the following:

First, free will is only granted to humans. Therefore, by continuing to fulfill the Lord's commands through performing one's duties, and by abandoning all attachments to those actions and their results, using them as a means for realizing God, a person can easily realize God. Arjuna, possessing a human body, enjoyed freedom of action. Therefore, he should have made the most of that opportunity.

Secondly, people only have the right to perform actions, but not the right to abandon them. If one tries to forcibly renounce all action from a sense of self, it will not succeed (III.5). This is because human nature drives individuals towards action (III.33, XVIII.59, 60). By doing so, one would be abusing their authority and incurring negative consequences by refusing to fulfill their duties, thereby disobeying the commandments in the scriptures. Therefore, Arjuna was obligated to perform his duty resolutely, and he should not have abandoned it.

Thirdly, just as the government permits humans to possess and use weapons and ammunition under certain rules, humans are also given a body with mind, intellect, and senses, freed from the constraints of worldly existence between life and death, and endowed with the power and authority to perform new actions for the benefit of others.

Those who make the best use of this opportunity will be liberated from the bonds of karma and attain the highest state. However, those who misuse this opportunity will be punished, and their authority will be taken away. In other words, they will be sent back to a lower level of creation. Knowing this secret, people should make the most of this opportunity.

The statement "People have no right to the fruits of their actions" is made to show that people cannot freely control the outcomes of their own actions.

A person does not know which action will produce what kind of result, when, how, or to what extent. Even if a person desires a certain outcome, they may not necessarily obtain it, and they cannot always avoid unwanted outcomes. The distribution of results is under the control of divine will, and people are completely powerless in that regard.

Regarding the rituals recommended in the scriptures, it is the same.

If a ritual is performed according to the regulations of the scripture in order to satisfy a particular deity, then that ritual is said to yield the desired results. However, even those results are not under the control of the person performing the ritual, but rather belong to the deity they were trying to appease. Therefore, it is ignorant to be attached to wealth, power, honor, fame, social prestige, worldly achievements, or heavenly attainments. All of these are temporary and unstable, and do not possess true stability. That is why Arjuna should not desire the fruits of any action.

Then, should we eliminate even the desire for redemption from our hearts? Desiring redemption is a noble wish and can be helpful in achieving it. However, unless one gains insight into God's truth and reality, it is difficult to detach oneself from even the subtle desires associated with wanting redemption. Furthermore, eliminating the desire for redemption does not necessarily mean that redemption will come sooner. Even desiring redemption is a form of desire, indirectly.

"Not to be the cause that produces the fruit of action" means that a person who uses their body, mind, and intellect to perform actions recognized in scripture, while harboring attachment, hope, and desire for the result, becomes the cause that brings about that result.

Only those who are attached to both the action and its result will receive that result. Those who have abandoned all attachment to both the action and its result will not receive it. Therefore, in order to attain ultimate peace, Arjuna should perform his duty while completely abandoning all attachments and desires.

The person who acts in such a way does not make their actions the cause of joy or suffering. All of his good and bad deeds lose the power to bear fruit. Attachment is the inducement for sinful behavior. When attachment and desire do not exist at all, a person will not commit new sins. Furthermore, past sins are burned away by the fire of present actions that have been separated from attachment and desire.

In this way, actions performed without attachment and desire are neutralized (IV.23), and the actor reaches a state of ultimate bliss (II.51).

Finally, the Lord is teaching Arjuna not to be attached to non-action.

Those who indulge in forbidden actions misuse their right to perform actions that are permitted by scriptures. Similarly, those who do not fulfill the duties prescribed according to one's status, stage of life, nature, and circumstances also misuse that right. Abandonment of duty cannot be justified. Abandonment due to ignorance is a abandonment of dharma (XVIII.7), and abandonment due to physical discomfort is a abandonment of rajas (XVIII.8). It is not possible to attain the fulfillment of karma yoga without performing one's prescribed duties (III.4). Therefore, the Lord advises Arjuna to avoid attachment to non-action and to certainly perform his prescribed duties.