Inter-relation between the four brahmaviharas?

Uppekkha is what is used to straighten Karuna. Not Muditha. Compassion can easily turn into sadness. When that happens you view the situation in terms of Karma and Vipaka: “All beings are the owners of their karma; their happiness and unhappiness depends upon their actions, not on my wishes for them.”

Metta(kindness) can turn into lust when practiced towards attractive women. Then you can use a mental technique like thinking of them as mothers, sisters, daughters or you can meditate on the repulsiveness of the body.

In the essay "The Four Sublime States" by Ven. Nyanaponika Thera, there is a chapter entitled "The Inter-relations of the Four Sublime States".

I quote a part of it below:

The translation used here: Love = metta, compassion = karuna, equanimity = upekkha, sympathetic joy = mudita. I would have preferred to translate mudita as "empathetic joy" rather than "sympathetic joy".

Unbounded love guards compassion against turning into partiality, prevents it from making discriminations by selecting and excluding and thus protects it from falling into partiality or aversion against the excluded side.

Love imparts to equanimity its selflessness, its boundless nature and even its fervor. For fervor, too, transformed and controlled, is part of perfect equanimity, strengthening its power of keen penetration and wise restraint.

Compassion prevents love and sympathetic joy from forgetting that, while both are enjoying or giving temporary and limited happiness, there still exist at that time most dreadful states of suffering in the world. It reminds them that their happiness coexists with measureless misery, perhaps at the next doorstep. It is a reminder to love and sympathetic joy that there is more suffering in the world than they are able to mitigate; that, after the effect of such mitigation has vanished, sorrow and pain are sure to arise anew until suffering is uprooted entirely at the attainment of Nibbana. Compassion does not allow that love and sympathetic joy shut themselves up against the wide world by confining themselves to a narrow sector of it. Compassion prevents love and sympathetic joy from turning into states of self-satisfied complacency within a jealously-guarded petty happiness. Compassion stirs and urges love to widen its sphere; it stirs and urges sympathetic joy to search for fresh nourishment. Thus it helps both of them to grow into truly boundless states (appamañña).

Compassion guards equanimity from falling into a cold indifference, and keeps it from indolent or selfish isolation. Until equanimity has reached perfection, compassion urges it to enter again and again the battle of the world, in order to be able to stand the test, by hardening and strengthening itself.

Sympathetic joy holds compassion back from becoming overwhelmed by the sight of the world's suffering, from being absorbed by it to the exclusion of everything else. Sympathetic joy relieves the tension of mind, soothes the painful burning of the compassionate heart. It keeps compassion away from melancholic brooding without purpose, from a futile sentimentality that merely weakens and consumes the strength of mind and heart. Sympathetic joy develops compassion into active sympathy.

Sympathetic joy gives to equanimity the mild serenity that softens its stern appearance. It is the divine smile on the face of the Enlightened One, a smile that persists in spite of his deep knowledge of the world's suffering, a smile that gives solace and hope, fearlessness and confidence: "Wide open are the doors to deliverance," thus it speaks.

Equanimity rooted in insight is the guiding and restraining power for the other three sublime states. It points out to them the direction they have to take, and sees to it that this direction is followed. Equanimity guards love and compassion from being dissipated in vain quests and from going astray in the labyrinths of uncontrolled emotion. Equanimity, being a vigilant self-control for the sake of the final goal, does not allow sympathetic joy to rest content with humble results, forgetting the real aims we have to strive for.

Equanimity, which means "even-mindedness," gives to love an even, unchanging firmness and loyalty. It endows it with the great virtue of patience. Equanimity furnishes compassion with an even, unwavering courage and fearlessness, enabling it to face the awesome abyss of misery and despair which confront boundless compassion again and again. To the active side of compassion, equanimity is the calm and firm hand led by wisdom — indispensable to those who want to practice the difficult art of helping others. And here again equanimity means patience, the patient devotion to the work of compassion.

In these and other ways equanimity may be said to be the crown and culmination of the other three sublime states. The first three, if unconnected with equanimity and insight, may dwindle away due to the lack of a stabilizing factor. Isolated virtues, if unsupported by other qualities which give them either the needed firmness or pliancy, often deteriorate into their own characteristic defects. For instance, loving-kindness, without energy and insight, may easily decline to a mere sentimental goodness of weak and unreliable nature. Moreover, such isolated virtues may often carry us in a direction contrary to our original aims and contrary to the welfare of others, too. It is the firm and balanced character of a person that knits isolated virtues into an organic and harmonious whole, within which the single qualities exhibit their best manifestations and avoid the pitfalls of their respective weaknesses. And this is the very function of equanimity, the way it contributes to an ideal relationship between all four sublime states.