# Lie Algebra: Optimal system of one-dimensional sub-algebras of the heat equation

This is a follow up question to Invariants of a PDE by Lie Symmetries, as I tried to follow the reasoning from the book Applications of Lie Groups to Differential Equations (Peter J. Olver, Example 3.13 on page 204-207).

As one can show the heat equation $u_t=u_{xx}$ has the following symmetries (infinitesimal transformations): $$v_1 = \partial_x \quad v_2 = \partial_t \quad v_3 = u\partial_u \quad v_4 = x\partial_x +2t\partial_t$$ $$v_5 = 2t\partial_x-xu\partial_u \quad v_6 = 4tx\partial_x+4t^2\partial_t-(x^2+2t)u\partial_u$$ $$v_{\alpha} = \alpha(x,t)\partial_u$$

Where $\alpha(x,t)$ is a solution of the heat equation.

In example 3.13 the author wants to derive the optimal system of sub-algebras of the heat equation.

Getting the commutator table and adjoint table is not a problem. But from there on everything is not clear to me.

The author starts with the general vector $$v=a_1v_1+a_2v_2+a_3v_3+a_4v_4+a_5v_5+a_6v_6$$

and directly concludes that $\eta(v)=a_4^2-4a_2a_6$ is an invariant of the full adjoint action: $\eta(Ad g(v))=\eta(v), v \in g, g \in G$

My first question: How am I supposed to find this invariant?

EDIT: I found out that this has to do with the killing form of the lie algebra.

Then it is stated that:

$$\tilde{v}=\sum_{i=1}^6\tilde{a}_iv_i=Ad(\exp(\alpha v_6))\circ Ad(\exp(\beta v_2))v.$$

My second question: Where does this come from?

He then continues to simplify and finally gets to the set of optimal subalgebras of the heat equation

My last question: Can someone explain what he does?

Thank you alot for reading my question :).