Almost primitive elements of free nonassociative algebras of small ranks

Let $K$ be a field, $X=\{x_1,\dots,x_n\}$, and let $F(X)$ be the free nonassociative algebra over the field $K$ with the set $X$ of free generators. A. G. Kurosh proved that subalgebras of free nonassociative algebras are free. A subset $M$ of nonzero elements of the algebra $F(X)$ is said to be primitive if there is a set $Y$ of free generators of $F(X)$, $F(X)=F(Y)$, such that $M\subseteq Y$ (in this case we have $|Y|=|X|=n$). A nonzero element $u$ of the free algebra $F(X)$ is said to be an almost primitive if $u$ is not a primitive element of the algebra $F(X)$, but $u$ is a primitive element of any proper subalgebra of $F(X)$ that contains it. In this article, for free nonassociative algebras of rank 1 and 2 criteria for homogeneous elements to be almost primitive are obtained and algorithms to recognize homogeneous almost primitive elements are constructed. New examples of almost primitive elements of free nonassociative algebras of rank 3 are constructed.