Algebra subgroup of algebra subgroup is algebra subgroup

Definition
Suppose $$G$$ is the algebra group corresponding to an associative algebra $$N$$ (that admits an algebra group) over a field $$F$$, i.e., $$G = 1 + N$$. Suppose $$K$$ is an algebra subgroup of $$G$$. Suppose $$H$$ is an algebra subgroup of $$K$$ with the algebra group structure that $$K$$ inherits. Then, $$H$$ is an algebra subgroup of $$G$$.