Corollary
Tag 09V3, permanent. Cite it as [Bourbaki, Tag 09V3].
Lie Groups and Lie Algebras, chapter I, § 2. Enveloping algebra of a Lie algebra
Suppose that K is a field and that g is finite-dimensional over K. Let I₁, ..., Iₘ be right (resp. left) ideals of finite codimension in U. Then the product ideal I₁I₂...Iₘ is of finite codimension.
By induction on m it suffices to consider the case of, for example, two right ideals. The right U-module I₁ is generated by a finite number of elements u₁, ..., uₚ (Proposition 6). Let v₁, ..., vₐ be elements of U whose classes modulo I₂ generate the vector space U/I₂. Then the canonical images in I₁/I₁I₂ of the uᵢvⱼ generate the vector space I₁/I₁I₂, which is therefore finite-dimensional. Hence dim_K(U/I₁I₂) = dim_K(U/I₁) + dim_K(I₁/I₁I₂) < +∞.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.