Alright. Leftism can mean many things, political liberalism, social liberalism, economic liberalism, etc. Socialism is very focused on economic liberalism, and through the seizure of economic power political liberalism. Your statement does not leave any leeway for people who might be politically liberal and yet want to maintain economic inequality (such as our Founding Fathers...a bunch of rich white men who didn't want to pay taxes.) Thus your statement is wrong. People can be leftists without being socialists.
I assume, in the first instance, that by "Liberalism" you mean "Liberalism" in the modern, post WWII meaning of the term, which is, in essence...Leftism. If this is the case, then I would say the following:
I do not believe that economic liberalism and social liberalism can be effectively extracted and separated from one another in either a theoretical or practical manner (I'm not sure about what you term "political liberalism", as it would seem that "political liberalism" would encompass the other two, politics having to do more broadly with the structure and organization of society as a whole). Social liberalism requires statism and a collectivizing coerciveness within the apparatus of the state to achieve most of its goals (homosexual marriage, outcome based education, etc.). At the same time, economic liberalism (usually some form of Marxian or Marx influenced socialism) implies and must impose a collectivist mentality and social frame of reference among the population such that the collectivizing of economic life (as this effects all other aspects of social life) can proceed without being encumbered by resistance generated by institutions and attitudes inimical to that collectivist mentality (the family, religion, etc.). Social liberalism then (the sexual revolution, radical feminism, the post King civil rights movement, the Gay rights movement, outcome based education, the assault in the public schools on competition and competitive sports, the denigration of excellence, achievement, and of "standing out" from the crowd; reductionistic, medicalized explanations for criminality, addiction, and other forms forms of anti-social behavior, the continuing war on boys, men, and masculinity (another long standing project of radical feminism), moral and value relativism etc.) cultivates the attitudes and value shifts necessary for economic liberalism, or, in other words, economic liberalism requires social liberalism to remove the moral, ethical, and psychological barriers among a critical mass of a people to a highly controlled, delimited, collectivist economic system. And since the control of economic life is, in essence, the control of all other aspects of life, "social" leftism appears as simply a necessary appendage to the means used by any political class or social movement seeking to impose a socialistic economic order, which is to say, a socialist social order.
"Political" leftism is simply a catch-all term that encompasses both economics and socio-cultural issues.
Socialism is very focused on economic liberalism, and through the seizure of economic power
The most effective of all the socialist movements of the 20the century, Cultural Marxism, sees socialism as a concept comprehending all of human life and activity, including media, art, music, and all forms of "cultural production". One cannot socialize a society economically without socializing that society qua the society. Marx knew this just as well as his later Frankfurt School disciples. The idea of the base and the superstructure; that all cultural artifacts and productions: music, art, literature, philosophy, religion, and values, are nothing but props for class interests grounded in economic relations between those classes is a case in point. Economic socialization must affect all other aspects of a society, and without altering those aspects, economic socialism will encounter strong resistance from traditional value structures embedded within the natural buffers against the state, primarily the family, church, and other private, voluntary organizations existing at a grass roots level.
Your statement does not leave any leeway for people who might be politically liberal and yet want to maintain economic inequality (such as our Founding Fathers...a bunch of rich white men who didn't want to pay taxes.) Thus your statement is wrong. People can be leftists without being socialists.
You appear to be engaging in some very gross equivocation of terms here, which will make further discussion difficult. In your first paragraph, you equate Socialism with something you call political "liberalism". Now here, you mention something called political "liberalism" and attach it to the Founders. But the founders were classical liberals, not "liberals" in the modern sense, and their political philosophy was, and is, the antithesis of Leftism. Which is which?
Further, all the Founders had to do to "maintain economic equality" was to create a government and constitution who's primary function was to protect and preserve liberty. Economic liberty under a rule of law will inevitably produce economic inequality. That is to say, freedom allows the natural distinctions in ability and aptitude among human beings to manifest themselves more fully and in greater profusion and variety.
Hence, the Founders didn't "maintain economic inequality" at all. They allowed freedom, property rights, and forbade direct taxation of individuals. In such a system, a vast number of levels of economic achievement will be manifest. They didn't "do" anything at all.