I was trying to prove that an operator $T$ in a real vector space $V$ has an upper block triangular matrix with each block being $1 \times 1$ or $2 \times 2$ and without using induction.
The procedure which i followed was :
We already know that an operator in a real vector space has either a one dimensional invariant subspace or a 2 dimensional invariant subspace.
Whatever be the case now, lets begin with the vector(s) which span these subspaces.
Let U denote this subspace ----- (1)
Now, if i am able to prove that there exists an another subspace W such that T is an invariant operator on the direct sum of U and W , then we can prove that operator T in a real vector space V has an upper block triangular matrix .
I need a direction on proving the latter part.
I sincerely thank you for the help.